Linear Differential Equations
Class 12th Mathematics RS Aggarwal Solution
- {dy}/{dx} + frac {1}/{x} c. y = x^{2} Find the general solution for each of…
- x {dy}/{dx} + 2y = x^{2} Find the general solution for each of the following…
- 2x {dy}/{dx} + y = 6x^{3} Find the general solution for each of the following…
- x {dy}/{dx} + y = 3x^{2} - 2 , x>0 Find the general solution for each of the…
- x {dy}/{dx} - y = 2x^{3} Find the general solution for each of the following…
- x {dy}/{dx} - y = x+1 Find the general solution for each of the following…
- ( 1+x^{2} ) {dy}/{dx} + 2xy = frac {1}/{ ( 1+x^{2} ) } Find the general…
- ( 1-x^{2} ) {dy}/{dx} + xy = x root { 1-x^{2} } Find the general solution…
- ( 1-x^{2} ) {dy}/{dx} + xy = ax Find the general solution for each of the…
- ( x^{2} + 1 ) {dy}/{dx} - 2xy = ( x^{2} + 1 ) ( x^{2} + 2 ) Find the general…
- {dy}/{dx} + 2y = 6e^{x} Find the general solution for each of the following…
- {dy}/{dx} + 3y = e^{-2x} Find the general solution for each of the following…
- {dy}/{dx} + 8y = 5e^{-3x} Find the general solution for each of the following…
- x {dy}/{dx} - y = (x-1) e^{x} , x>0 Find the general solution for each of…
- {dy}/{dx} - ytanx = e^{x}secx Find the general solution for each of the…
- (xlogx) {dy}/{dx} + y = 2logx Find the general solution for each of the…
- x {dy}/{dx} + y = xlogx Find the general solution for each of the following…
- x {dy}/{dx} + 2y = x^{2}logx Find the general solution for each of the…
- (1+x) {dy}/{dx} - y = e^{3x} (1+x)^{2} Find the general solution for each of…
- {dy}/{dx} + frac {4x}/{ ( x^{2} + 1 ) } y + frac {1}/{ ( x^{2} + 1 ) ^{2} } =…
- ( y+3x^{2} ) {dx}/{dy} = x Find the general solution for each of the…
- xdy - ( y+2x^{2} ) dx = 0 Find the general solution for each of the following…
- xdy + ( y-x^{3} ) dx = 0 Find the general solution for each of the following…
- {dy}/{dx} + 2y = sinx Find the general solution for each of the following…
- {dy}/{dx} + y = cosx-sinx Find the general solution for each of the following…
- secx {dy}/{dx} - y = sinx Find the general solution for each of the…
- ( 1+x^{2} ) {dy}/{dx} + 2xy = cotx Find the general solution for each of the…
- (sinx) {dy}/{dx} + (cosx) y = cosxsin^{2}x Find the general solution for…
- {dy}/{dx} + 2ycotx = 3x^{2}cosec^{2}x Find the general solution for each of…
- x {dy}/{dx} - y = 2x^{2}secx Find the general solution for each of the…
- {dy}/{dx} = ytanx-2sinx Find the general solution for each of the following…
- {dy}/{dx} = ycotx = sin2x Find the general solution for each of the following…
- {dy}/{dx} + 2ytanx = sinx Find the general solution for each of the following…
- {dy}/{dx} + ycotx = x^{2}cotx+2x Find the general solution for each of the…
- x {dy}/{dx} + y = x^{3} , given that 𝒴 =1 when 𝒳 =2 Find a particular…
- {dy}/{dx} + ycotx = 4xcosecx , given that 𝒴 = 0 when 𝒳 = { pi }/{2} .…
- {dy}/{dx} + 2xy = x , given that 𝒴 = 0 when 𝒳 =0. Find a particular solution…
- {dy}/{dx} + 2y = e^{-2x}sinx , given that 𝒴 = 0, when 𝒳 = 0. Find a…
- ( 1+x^{2} ) {dy}/{dx} + 2xy = 4x^{2} , given that 𝒴 = 0 when 𝒳 = 0. Find…
- x {dy}/{dx} - y = logx , given that 𝒴 = 0 when 𝒳 = 1. Find a particular…
- {dy}/{dx} + ytanx = 2x+x^{2}tanx , given that 𝒴 = 1 when 𝒳 = 0. Find a…
- A curve passes through the origin and the slope of the tangent to the curve at…
- A curve passes through the point (0, 2) and the sum of the coordinates of any…
- ydx - ( x+2y^{2} ) dy = 0 Find the general solution for each of the following…
- ydx + ( x-y^{2} ) dy = 0 Find the general solution for each of the following…
- ydx + ( x-y^{2} ) dy = 0 Find the general solution for each of the following…
- ( x+3y^{3} ) {dy}/{dx} = y , (y>0) Find the general solution for each of the…
- (x+y) {dy}/{dx} = 1 Find the general solution for each of the following…
- (x+y+1) {dy}/{dx} = 1 Find the general solution for each of the following…
- Solve (x+1) {dy}/{dx} = 2e^{-y} - 1 , given that 𝒳 = 0 when 𝒴 = 0.…
- Solve ( 1+y^{2} ) dx + (x-e^{-tan^{-1}y}) dy = 0 , given that when 𝒴 =o,…
- The solution of the de {dy}/{dx} = e^{x+y} is Mark (√) against the correct answer…
- The solution of the de {dy}/{dx} = 2^{x+y} is Mark (√) against the correct answer…
- The solution of the de ( e^{x} + 1 ) y dy = (y+1) e^{x} dx is Mark (√) against the…
- The solution of the dexdy+ydx = 0 is Mark (√) against the correct answer in the…
- The solution of the is Mark (√) against the correct answer in the following:…
- The solution of the de {dy}/{dx} = frac { ( 1+y^{2} ) }/{ ( 1+x^{2} ) } is. Mark…
- The solution of the de {dy}/{dx} =1 - 𝒳 + 𝒴 – 𝒳𝒴 is Mark (√) against the…
- The solution of the de {dy}/{dx} = e^{x+y} + x^{2} c. e^{y} is Mark (√) against…
- The solution of the de {dy}/{dx} + root { frac { 1-y^{2} }/{ 1-x^{2} } } = 0 is…
- The solution of the de {dy}/{dx} = frac {1-cosx}/{1+cosx} is Mark (√) against the…
- The solution of the {dy}/{dx} = frac {-2xy}/{ ( x^{2} + 1 ) } is Mark (√) against…
- The solution of the DE cos 𝒳 (1 + cos 𝒴 ) 𝒹𝒳 – sin 𝒴 (1 + sin 𝒳) 𝒹𝒴 = 0 is…
- the solution of the DE 𝒳 cos 𝒴 𝒹𝒴 = (𝒳e𝒳 log 𝒳 + e𝒳 ) 𝒹𝒳 is Mark (√) against…
- The solution of the DE {dy}/{dx} + 𝒴 log 𝒴 cot 𝒳 = 0 is Mark (√) against the…
- the general solution of the DE (1 + 𝒳2) 𝒹𝒴 – 𝒳𝒴 𝒹𝒳 = 0 is Mark (√) against the…
- The general solution of the dex root { 1+y^{2} } dx+y sqrt { 1+x^{2} } dy =0 is Mark…
- The general solution of the DE log ( {dy}/{dx} ) = (ax+by) is Mark (√) against the…
- The general solution of the de {dy}/{dx} = ( root { 1-x^{2} } ) ( sqrt { 1-y^{2} }…
- The general solution of the DE {dy}/{dx} = frac { y^{2} - x^{2} }/{2xy} is Mark…
- The general solution of the DE x^{2} {dy}/{dx} = x^{2} + xy+y^{2} is. Mark (√)…
- The general solution od the DE {dy}/{dx} = y+xtan frac {y}/{x} is Mark (√)…
- The general solution of the DE 2𝒳𝒴 𝒹𝒴 + (𝒳2 - 𝒴2) 𝒹𝒳 = 0 is Mark (√) against…
- The general solution of the DE (𝒳 - 𝒴 ) 𝒹𝒴 + (𝒳 + 𝒴) 𝒹𝒳 is Mark (√) against…
- The general solution of the DE {dy}/{dx} = frac {y}/{x} + sin frac {y}/{x} is…
- The general solution of the DE {dy}/{dx} + ytanx = secx is Mark (√) against the…
- The general solution of the DE {dy}/{dx} + ycotx = 2cosx is Mark (√) against the…
- The general solution of the DE {dy}/{dx} + frac {y}/{x} = x^{2} is Mark (√)…
Exercise 21
Question 1.Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
………eq(1)
Formula :
i) 
ii) 
iii) 
iv) General solution :
For the differential equation in the form of
The general solution is given by,
Where integrating factor,
Answer :
Equation (1) is of the form
Where, 
and Q = x2
Therefore, integrating factor is
………
= x ………
General solution is
………
Question 2.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii) 
iv)
v) General solution :
For the differential equation in the form of
The general solution is given by,
Where integrating factor,
Answer :
Given differential equation is
Dividing the above equation by x,
………eq(1)
Equation (1) is of the form
Where, 
and Q = x
Therefore, integrating factor is
………
………
= x2 ………
General solution is
………
Question 3.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
The general solution is given by,
Where integrating factor,
Answer :
Given differential equation is
Dividing the above equation by 2x,
………eq(1)
Equation (1) is of the form
Where, 
and Q = 3x2
Therefore, integrating factor is
………
………
= √x………
General solution is
………
Dividing the above equation by √x
Question 4.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) General solution :
For the differential equation in the form of
The general solution is given by,
Where integrating factor,
Answer :
Given differential equation is
Dividing the above equation by x,
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, the integrating factor is
………
= x………
General solution is
………
Dividing the above equation by x
Question 5.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
The general solution is given by,
Where integrating factor,
Answer :
Given differential equation is
Dividing the above equation by x,
………eq(1)
Equation (1) is of the form
Where, 
and Q = 2x2
Therefore, integrating factor is
………
………
………
General solution is
………
Multiplying above equation by x
Question 6.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by x,
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
………
General solution is
………
Multiplying above equation by x,
Question 7.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii) 
iii)
iv) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by (1+x2),
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
Let, 
………
………
General solution is
………
Therefore, general solution is
Question 8.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii) 
iii) 
iv) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by (1 – x2),
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
Let (1 – x2) = f(x)
Therefore f’(x) = -2x
……eq(2)
………
………
General solution is
………from eq(2)
Multiplying above equation by 
,
Question 9.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii)
iii)
iv) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by (1 – x2),
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
Let (1 – x2) = f(x)
Therefore f’(x) = -2x
………
………
General solution is
……eq(2)
Let
Put (1 – x2) = t
Substituting I in eq(2)
Multiplying above equation by 
,
Question 10.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) 
v) 
vi) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by (1 + x2),
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
Let (1 + x2) = f(x)
Therefore f’(x) = 2x
………
………
General solution is
………
Therefore general solution is
Question 11.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii) 
iii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
General solution is
………
Dividing above equation by 
,
Therefore general solution is
Question 12.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
General solution is
………
Dividing above equation by 
,
Therefore general solution is
Question 13.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
General solution is
………
Dividing above equation by 
,
Therefore general solution is
Question 14.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) 
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by x,
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
………
General solution is
………eq(2)
Let,
Let 
………
Substituting I in eq(2),
Multiplying above equation by x,
Therefore general solution is
Question 15.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii)
iii)
iv) 
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
………
………
General solution is
………
Therefore general solution is
Question 16.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii)
iii) 
iv) 
v) 
vi) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by (x.log x),
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
Let, 
………
………
General solution is
………eq(2)
Let,
Let, 
………
………
Substituting I in eq(2),
Question 17.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii)
iii)
iv)
v) 
vi) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by x,
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
General solution is
………eq(2)
Let,
Let, 
………
………
………
Substituting I in eq(2),
Multiplying above equation by 4,
Therefore general equation is
Question 18.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii) 
iii)
iv)
v)
vi)
vii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by x,
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
………
General solution is
………eq(2)
Let,
Let, 
………
………
………
Substituting I in eq(2),
Dividing above equation by x2,
Therefore general equation is
Question 19.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii)
iii)
iv) 
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by (1+x),
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
………
………
General solution is
………
Multiplying above equation by (1+x),
Therefore general equation is
Question 20.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii)
iii)
iv) 
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
Let, 
………
………
………
General solution is
………
Dividing above equation by (1+x2)2,
Therefore general equation is
Question 21.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
………
General solution is
………
Multiplying above equation by x,
Therefore general equation is
Question 22.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
………
General solution is
………
Multiplying above equation by x,
Therefore general equation is
Question 23.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
General solution is
………
Dividing above equation by x,
Therefore general equation is
Question 24.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii) 
iii) 
iv) 
v) 
vi) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
General solution is
………eq(2)
Let,
Let, u=sin x and v=e2x
………
………
Again, let u=cos x and v=e2x
………
………
Multiplying above equation by 4,
Substituting I in eq(2),
Dividing above equation by e2x,
Therefore general equation is
Question 25.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii) 
iii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
General solution is
Let, f(x)=cos x => f ’(x) = -sin x
………
Dividing above equation by ex,
Therefore general equation is
Question 26.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii) 
iii)
iv) 
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by sec x,
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
General solution is
………eq(2)
Let,
Put sin x=t => cos x.dx=dt
………
………
………
Substituting I in eq(2),
Dividing above equation by e-sinx,
Therefore general equation is
Question 27.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii)
iii) 
iv) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by (1+x2),
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
Let, f(x) = (1+x2) => f ’ (x) = 2x
………
………
General solution is
………
Therefore, general solution is
Question 28.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
v) 
vi)
vii)
viii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by sin x,
………eq(1)
Equation (1) is of the form
Where, 
and
Therefore, integrating factor is
………
= sin x ………
General solution is
………eq(2)
Let,
Put sin x=t => cos x.dx=dt
………
Substituting I in eq(2),
Therefore, general solution is
Question 29.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
………
= sin2 x ………
General solution is
………
Therefore, general solution is
Question 30.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
vi)
vii)
viii)
ix)
x) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Dividing above equation by x,
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
………
General solution is
………
Multiplying above equation by x,
Therefore, general solution is
Question 31.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii)
iii)
iv) 
v) 
vi) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
………
General solution is
………
………
Multiplying above equation by 2,
where, C=2c
Therefore, general solution is
Question 32.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i) 
ii)
iii)
iv)
v)
vi)
vii) 
viii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
General solution is
………eq(2)
Let,
Let, u=sin 2x & v=sin x
………
………
Again let, u=cos 2x & v=cos x
………
………
Substituting I in eq(2),
Therefore, general solution is
Question 33.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) 
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and
Therefore, integrating factor is
………
………
………
General solution is
………eq(2)
Let,
Put, cos x=t => -sin x dx = dt
………
Substituting I in eq(2),
Multiplying above equation by cos2x,
Therefore, general solution is
Question 34.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) 
vi) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
= sin x ………
General solution is
………eq(2)
Let,
Let, u=x2 and v=cos x
………
………
Substituting I in eq(2),
Dividing above equation by sin x,
Therefore, general solution is
Question 35.
Find a particular solution satisfying the given condition for each of the following differential equations.
, given that 𝒴 =1 when 𝒳 =2
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by x,
………eq(1)
Equation (1) is of the form
Where, and
Therefore, integrating factor is
………
………
General solution is
………
Dividing above equation by x,
Therefore general equation is
For particular solution put y=1 and x=2 in above equation,
Therefore, particular solution is
Question 36.
Find a particular solution satisfying the given condition for each of the following differential equations.
, given that 𝒴 = 0 when 𝒳 =
.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
= sin x ………
General solution is
………
Therefore general equation is
For particular solution put y=0 and 
in above equation,
Therefore, particular solution is
Question 37.
Find a particular solution satisfying the given condition for each of the following differential equations.
, given that 𝒴 = 0 when 𝒳 =0.
Answer:
Given Differential Equation :
Formula :
i)
ii) 
iii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
General solution is
………eq(2)
Let,
Put, x2=t => 2x dx = dt
………
Substituting I in eq(2),
Therefore, general solution is
For particular solution put y=0 and x=0 in above equation,
Substituting c in general solution,
Multiplying above equation by 
Therefore, particular solution is
Question 38.
Find a particular solution satisfying the given condition for each of the following differential equations.
, given that 𝒴 = 0, when 𝒳 = 0.
Answer:
Given Differential Equation :
Formula :
i)
ii) 
iii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
General solution is
………
Therefore, general solution is
For particular solution put y=0 and x=0 in above equation,
Substituting c in general solution,
Therefore, particular solution is
Question 39.
Find a particular solution satisfying the given condition for each of the following differential equations.
, given that 𝒴 = 0 when 𝒳 = 0.
Answer:
Given Differential Equation :
Formula :
i) 
ii) 
iii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by (1+x2),
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
Let, 
………
General solution is
………
Therefore, general solution is
For particular solution put y=0 and x=0 in above equation,
Substituting c in general solution,
Dividing above equation by (1+x2),
Therefore, particular solution is
Question 40.
Find a particular solution satisfying the given condition for each of the following differential equations.
, given that 𝒴 = 0 when 𝒳 = 1.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) 
v) 
vi)
vii) 
viii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
Dividing above equation by x,
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
………
General solution is
………eq(2)
Let,
Put, log x =t => x=et
Therefore, (1/x) dx = dt
Let, u=t and v=e-t
………
………
………
Substituting I in eq(2),
Multiplying above equation by x,
Therefore, general solution is
For particular solution put y=0 and x=1 in above equation,
………
Substituting c in general solution,
Therefore, particular solution is
Question 41.
Find a particular solution satisfying the given condition for each of the following differential equations.
, given that 𝒴 = 1 when 𝒳 = 0.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) 
v)
vi) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
………
General solution is
………eq(2)
Let,
Let, u=x2 and v= tan x. sec x
………
………
Substituting I in eq(2),
Multiplying above equation by cos x,
Therefore, general solution is
For particular solution put y=1 and x=0 in above equation,
Substituting c in general solution,
Therefore, particular solution is
Question 42.
A curve passes through the origin and the slope of the tangent to the curve at any point (𝒳, ) is equal to the sum of the coordinates of the point. Find the equation of the curve.
Answer:
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
The slope of the tangent to the curve 
The slope of the tangent to the curve is equal to the sum of the coordinates of the point.
Therefore differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and
Therefore, integrating factor is
………
General solution is
………eq(2)
Let,
Let, u=x and v= e-x
………
………
………
Substituting I in eq(2),
Dividing above equation by e-x,
Therefore, general solution is
The curve passes through origin , therefore the above equation satisfies for x=0 and y=0,
Substituting c in general solution,
Therefore, equation of the curve is
Question 43.
A curve passes through the point (0, 2) and the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5. Find the equation of the curve.
Answer:
Formula :
i) 
ii)
iii)
iv) 
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
The slope of the tangent to the curve 
The sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at the given point by 5.
Therefore differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
General solution is
………eq(2)
Let,
Let, u=x-5 and v= e-x
………
………
………
Substituting I in eq(2),
Dividing above equation by e-x,
Therefore, general solution is
The curve passes through point (0,2) , therefore the above equation satisfies for x=0 and y=2,
Substituting c in general solution,
Therefore, equation of the curve is
Question 44.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
………
………
General solution is
………
Multiplying above equation by y,
Therefore, general solution is
Question 45.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
………
General solution is
………
Dividing above equation by y,
Therefore, general solution is
Question 46.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and
Therefore, integrating factor is
………
………
General solution is
………
Dividing above equation by y,
Therefore, general solution is
Question 47.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv) 
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
………
………
General solution is
………
Multiplying above equation by y,
Therefore, general solution is
Question 48.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and
Therefore, integrating factor is
………
General solution is
………eq(2)
Let,
Let, u=y and v= e-y
………
………
………
Substituting I in eq(2),
Therefore, general solution is
Question 49.
Find the general solution for each of the following differential equations.
Answer:
Given Differential Equation :
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, and 
Therefore, integrating factor is
………
General solution is
………eq(2)
Let,
Let, u=y+1 and v= e-y
………
………
………
Substituting I in eq(2),
Dividing above equation by e-y
Therefore, general solution is
Question 50.
Solve 
, given that 𝒳 = 0 when 𝒴 = 0.
Answer:
Given Equation:
Re-arranging, we get,
Let 2 – ey = t
-eydy = dt
Therefore,
Integrating both sides, we get,
log t = log(x + 1) + C
log (2 – ey) = log (x + 1) + C
At x = 0, y = 0.
Therefore,
log(2) = log(1) + C
Therefore,
C = log 2
Now, we have,
log (2 – ey) – log (x + 1) – log 2 = 0
Question 51.
Solve 
, given that when 𝒴 =o, then 𝒳 = 0.
Answer:
Given Differential Equation :
Formula :
i)
ii) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
Given differential equation is
………eq(1)
Equation (1) is of the form
Where, 
and 
Therefore, integrating factor is
………
General solution is
………
Putting x=0 and y=0
Therefore, general solution is
Objective Questions
Question 1.Mark (√) against the correct answer in the following:
The solution of the 
is
A. 
B. 
C. 
D. None of these
Answer:
Given, 
On integrating on both sides, we get
Conclusion: Therefore, is the solution of
Question 2.
Mark (√) against the correct answer in the following:
The solution of the 
is
A. 
B. 
C. 
D. None of these
Answer:
Given, 
On integrating on both sides, we get
Conclusion: Therefore, is the solution of
Question 3.
Mark (√) against the correct answer in the following:
The solution of the 
is
A. 
B. 
C. 
D. None of these
Answer:
Let, 
On differentiating on both sides we get 
Now we can write this equation as 
On integrating on both sides, we get
Conclusion: Therefore, is the solution of
Question 4.
Mark (√) against the correct answer in the following:
The solution of the 
is
A. 
B. 
C. 
D. None of these
Answer:
Given xdy + ydx = 0
xdy = -ydx
On integrating on both sides we get,
-log y = log x + c
log x + log y = c
log xy = c
xy = C
Conclusion: Therefore xy = c is the solution of xdy + ydx = 0
Question 5.
Mark (√) against the correct answer in the following:
The solution of the 
is
A. 
B. 
C. 
D. None of these
Answer:
Given:
Separating the variables, we get,
Integrating both sides, we get,
log sec y = log x + log c
xcosy = c
Hence, A is the correct answer.
Question 6.
Mark (√) against the correct answer in the following:
The solution of the 
is.
A. (𝒴 +𝒳 )=C(1-𝒴𝒳)
B. (𝒴 – 𝒳 ) = C(1+𝒴𝒳)
C. 𝒴 = (1+𝒳 )C
D. None of these
Answer:
Given 
On integrating on both sides, we get
(since 
)
y-x = C(1+yx)
Conclusion: Therefore, y-x = C(1+yx) is the solution of
Question 7.
Mark (√) against the correct answer in the following:
The solution of the 
=1 - 𝒳 + 𝒴 – 𝒳𝒴 is
A. Log (1 + 𝒴) = 𝒳 - 
+C
B. 
C. 
D. None of these
Answer:
On integrating on both sides, we get
Conclusion: Therefore, 
is the
solution of
Question 8.
Mark (√) against the correct answer in the following:
The solution of the 
is
A. 
B. 
C. 
D. None of these
Answer:
Given 
On integrating on both sides, we get
Conclusion: Therefore, 
is the
solution of
Question 9.
Mark (√) against the correct answer in the following:
The solution of the 
is
A. 𝒴 + sin-1𝒴 = sin-1𝒳 + C
B. sin-1𝒴 – sin-1𝒳 = C
C. sin-1𝒴 + sin-1𝒳 = C
D. None of these
Answer:
Given 
On integrating on both sides, we get
( As 
)
Conclusion: Therefore, is the
solution of
Question 10.
Mark (√) against the correct answer in the following:
The solution of the 
is
A. 
B. 
C. 
D. None of these
Answer:
Given 
On integrating on both sides, we get
Conclusion: Therefore, 
is the solution
of
Question 11.
Mark (√) against the correct answer in the following:
The solution of the 
is
A. 𝒴2 (𝒳 + 1) = C
B. 𝒴 (𝒳2 + 1) = C
C. 𝒳2 (𝒴 + 1) = C
D. None of these
Answer:
Given 
Let 
On differentiating on both sides we get 2xdx = dt
On integrating on both sides, we get
yt = C
As 
Conclusion: Therefore, is the solution of
Question 12.
Mark (√) against the correct answer in the following:
The solution of the DE cos 𝒳 (1 + cos 𝒴 ) 𝒹𝒳 – sin 𝒴 (1 + sin 𝒳) 𝒹𝒴 = 0 is
A. 1 + sin 𝒳 cos 𝒴 = C
B. (1 + sin 𝒳) (1 + cos 𝒴) = C
C. sin 𝒳 cos 𝒴 + cos 𝒳 = C
D. none of these
Answer:
Given cos x (1+cos y) dx – sin y (1+sin x) dy = 0
Let 1+cos y = t and 1+sin x = u
On differentiating both equations, we get
-sin y dy = dt and cos x dx = du
Substitute this in the first equation
t du + u dt = 0
-log u = log t + C
log u + log t = C
log ut = C
ut = C
(1+sin x)(1+cos y) = C
Conclusion: Therefore, (1+sin x)(1+cos y) = C is the solution of cos x (1+cos y) dx – sin y (1+sin x) dy = 0
Question 13.
Mark (√) against the correct answer in the following:
the solution of the DE 𝒳 cos 𝒴 𝒹𝒴 = (𝒳e𝒳 log 𝒳 + e𝒳 ) 𝒹𝒳 is
A. sin 𝒴 = e𝒳 log 𝒳 +C
B. sin 𝒴 - e𝒳 log 𝒳 = C
C. sin 𝒴 = e𝒳 (log 𝒳) +C
D. none of these
Answer:
Given 
On integrating on both sides we get
Conclusion: Therefore, the solution of
Question 14.
Mark (√) against the correct answer in the following:
The solution of the DE 
+ 𝒴 log 𝒴 cot 𝒳 = 0 is
A. cos 𝒳 log 𝒴 = C
B. sin 𝒳 log 𝒴 = C
C. log 𝒴 = C sin 𝒳
D. none of these
Answer:
Given 
Let log y = t
On differentiating we get
log t = -log (sin x) + C
log t + log(sin x) = C
log(tsin x) = C
tsin x = C
(log y)(sin x) = C
Conclusion: Therefore, (log y)(sin x) = C is the solution of
Question 15.
Mark (√) against the correct answer in the following:
the general solution of the DE (1 + 𝒳2) 𝒹𝒴 – 𝒳𝒴 𝒹𝒳 = 0 is
A. 𝒴 = C(1 + 𝒳2)
B. 𝒴2 = C(1 + 𝒳2)
C. 
= C
D. None of these
Answer:
Given 
Let 
2x dx = dt
On integrating on both sides we get
2 log y = log t + C
Conclusion: Therefore, is the solution of
Question 16.
Mark (√) against the correct answer in the following:
The general solution of the 
=0 is
A. sin-1𝒳 + sin-1𝒴 = C
B. 
C. tan-1𝒳 + tan-1𝒴 = C
D. None of these
Answer:
Given 
Let 
and 
2y dy = dt and 2x dx = du
On integrating on both sides we get
Conclusion: Therefore, 
is the
solution of
Question 17.
Mark (√) against the correct answer in the following:
The general solution of the DE 
is
A. 
B. 
C. 
D. None of these
Answer:
Given 
On integrating on both sides we get
Conclusion: Therefore, 
is the solution of
Question 18.
Mark (√) against the correct answer in the following:
The general solution of the 
is
A. 
B. 
C. 
D. None of these
Answer:
Given 
Let x = sin t
dx = cos t dt
We know 
On integrating on both sides we get
Sin 2t = 2 sin t cost
= 2x
Conclusion: Therefore, 
is the solution of
Question 19.
Mark (√) against the correct answer in the following:
The general solution of the DE 
is
A. 
B. 
C. 
D. None of these
Answer:
Given 
Let y = vx
On integrating on both sides, we get
Conclusion: Therefore, is the solution of
Question 20.
Mark (√) against the correct answer in the following:
The general solution of the DE 
is.
A. 
B. 
C. 
D. None of these
Answer:
Given 
Let y = vx
On integrating on both sides, we get
Conclusion: Therefore, 
is the solution of
Question 21.
Mark (√) against the correct answer in the following:
The general solution od the DE 
is
A. 
B. 
C. 
D. None of these
Answer: Given DE:
Now,
Dividing both sides by x, we get,
Let y = vx
Differentiating both sides,
dy/dx = v + xdv/dx
Now, our differential equation becomes,
On separating the variables, we get,
Integrating both sides, we get,
sinv = Cx
Putting the value of v we get,
Hence, B is the correct answer.
Question 22.
Mark (√) against the correct answer in the following:
The general solution of the DE 2𝒳𝒴 𝒹𝒴 + (𝒳2 - 𝒴2) 𝒹𝒳 = 0 is
A. 𝒳2 + 𝒴2 = C𝒳
B. 𝒳2 + 𝒴2 = C𝒴
C. 𝒳2 + 𝒴2 = C
D. None of these
Answer:
Given 
Let y = vx
On integrating on both sides, we get
Conclusion: Therefore, is the solution of
Question 23.
Mark (√) against the correct answer in the following:
The general solution of the DE (𝒳 - 𝒴 ) 𝒹𝒴 + (𝒳 + 𝒴) 𝒹𝒳 is
A. 
B. 
C. 
D. None of these
Answer:
Given (x-y)dy + (x+y) dx = 0
Let y = vx
Question is wrong. I think subtraction should be there instead of addition in LHS(left hand side)
Question 24.
Mark (√) against the correct answer in the following:
The general solution of the DE 
is
A. 
B. 
C. 
D. None of these
Answer:
Given 
Let y = vx
Conclusion: Therefore, 
is the solution of 
Question 25.
Mark (√) against the correct answer in the following:
The general solution of the DE 
is
A. 𝒴 = sin 𝒳 – C cos 𝒳
B. 𝒴 = sin 𝒳 + C cos 𝒳
C. 𝒴 = cos 𝒳 – C sin 𝒳
D. None of these
Answer:
Given 
It is in the form 
Integrating factor 
General solution 
y sec x = tan x + C
y = sin x + C cos x
Conclusion: Therefore, y = sin x + C cos x is the solution of
Question 26.
Mark (√) against the correct answer in the following:
The general solution of the DE 
is
A. (𝒴 + sin 𝒳)sin 𝒳 = C
B. (𝒴 + cos 𝒳 ) sin 𝒳 = C
C. (𝒴 – sin 𝒳 ) sin 𝒳 = C
D. None of these
Answer:
Given 
It is in the form
Integrating factor 
General solution is 
(y-sin x)sin x = C
Conclusion: Therefore, (y-sin x)sin x = C is the solution of
Question 27.
Mark (√) against the correct answer in the following:
The general solution of the DE 
is
A. 𝒳𝒴 = 𝒳4 + C
B. 4𝒳𝒴 = 𝒳4 + C
C. 3𝒳𝒴 = 𝒳3 + C
D. None of these
Answer:
Given 
It is in the form
Integrating factor 
General solution is 
Conclusion: Therefore, 
is the solution of