Differential Equations

QUESTION: 1

The solution of the differential equation yy' + y² - x = 0, where c is a constant, is

A.
y² = x + ce^-2x
B.
C.
D.
y² = a + ce^-2x + 1

Solution:

QUESTION: 2

Let k be a real constant. The solution of the differential equations satisfies the relation

A.
y - z = ke^3x
B.
y + y = ke^-x
C.
3y + z = ke^3x
D.
3y - z = ke^-3x

Solution:

Differentiating equation (i) w.r.t. we get

QUESTION: 3

If general solution of the differential equation ay",+ by"- cy’+ dy = 0 is linearly spanned by e^x sin x and cos x, then which one of the following holds?

A.
a + b - c - d = 0
B.
a + b + c + d = 0
C.
a - b + c - d = 0
D.
a - b - c - d = 0

Solution:

Linearly independent solutions of ay" + by”+cy' + dy = 0 are e^x cos x and sin x, so roots of aD³ + bD² + cD + d = 0 will be 1 and ±f, so we have 1 +

QUESTION: 4

Two linearly independent solutions of the differential equation y" - 2y' + y = 0 are y₁ = e^x and y₂ = xe^x. Then a particular solution of y² - 2y' + y + e^x sin x is

A.
y₁ cos x + y₂ (sin x - x cos x)
B.
y₁ sin x + y₂ (x cos x - sin x)
C.
y₁ (x cos x - sin x) - y₂ cos x
D.
y₁ (x sin x - cos x) + y₂ cos x

Solution:

Particular integral of y" - 2y' + y - e^x sin x is

QUESTION: 5

Orthogonal trajectories of the family of curves (x - 1)² + y² + 2ax = 0 are the solutions of the differential equation

Solution:

By putting the value of a from equation (ii) in equation (i), we get.

It ’s orthogonal trajectory will be D.E.

QUESTION: 6

Which one of the following differential equations represents all circles with radius a?

Solution:

Equation of the circles with radius ‘a' is

Putting values of (y - k) and (x - h) from equation (iv) and equation (v) in equation (i), we get

QUESTION: 7

The solution of the differential equation with initial condition y(0) = 0 is

A.
B.
C.
D.
x = y

Solution:

QUESTION: 8

The differential equation (2a² + by²) dx + cxydy = 0 is made exact by multiplying the integrating factor Then the relation between b and c is.

A.
2c = 0
B.
b = c
C.
2b + c = 0
D.
b + 2c = 0

Solution:

Integrating factor of (2x² + by²) dx + cxy dy = 0 is

QUESTION: 9

Given that Which one of the following is always true?

Solution:

....(i)
Differentiating both sides w.r.t. x, we get.

QUESTION: 10

The sum of the intercepts made on the axes of co-ordinates by any tangent to the curve √x + √y = 2 is equal to

A.
4
B.
8
C.
2
D.
none

Solution:

Equation of tangent is (Y - y) = y'(X - x)

Intercept on Y-axis.

Intercept on X-axis:

QUESTION: 11

Solution of the equation

A.
(x + a) (1 - ay) = cy
B.
(x + a) (1 + ay) = cy
C.
(1 + ax) (1 + y) = cy
D.
(1 - ax) (1 + y) = cy

Solution:

QUESTION: 12

If and it is known that for a = 1, y = 1; if x = -1, then the value of y will be :

A.
e²
B.
e
C.
1
D.
-1

Solution:

QUESTION: 13

The solution of the differential equation

A.
4x + y + 1 = tan (2x + c)
B.
4x + y + 1 = 2 tan (2x + c)
C.
2 (4x + y + 1) = tan (2x + c)
D.
tan (4a + y + 1) = 2a + c

Solution:

QUESTION: 14

If the solution of the differential equation is x + y -1 = Ce^u then the value of u is :

A.
x + y
B.
xy
C.
x + y -1
D.
x + y + 1

Solution:

Correct Answer :- C

Explanation : dy/dx = (x+y-z)/(x+y)

Let t = x + y

1 + dy/dx = dt/dx

Equation becomes,

dt/dx - 1 = (t-z)/t

dt/dx = (2t-z)/t

tdt/(2t-z) = dx

= 1/2[1+z/(2t-z)]dt = dx

t + 1/2log(2t-z) = 2x+c

2t-z = e2^(x-y+c)

x+y-1 = ce^x-y

QUESTION: 15

The solution of the equation

A.
s = t + Ce^-t
B.
s = t - 1 + Ce^-t
C.
s + t = Ce^-t
D.
s + t = Ce^-t - 1

Solution:

which is linear differential equation so, it’s solution is

QUESTION: 16

The solution of the differential equation

A.
e tan^-1 y = x tan^-1y + c
B.
xe^tan-1 y = tan^-1 y + c
C.
ye^tan-1 y = tan^-1 y + c
D.
ye^tan-1 y = y tan^-1 y + c

Solution:

which is linear differential equation in dependent variable x and independent variable z.
whose solution is

QUESTION: 17

The solution of the equation

A.
x = y(y² + c)
B.
xy = y² + c
C.
y = x(y² + c)
D.
x + y³ = c

Solution:

.......(i)
which is linear differential equation, so solution is

QUESTION: 18

The degree and order of differential equation are respectively

A.
order 2, degree 3
B.
order 1, degree 3
C.
order 3, degree 2
D.
order 3, degree 1

Solution:

QUESTION: 19

The solution of the equation (2x + y + 1)dx + (4x + 2y – 1)dy = 0 is:

A.
log (2x + y - 1) = C + x + y
B.
log (4x + 2y - 1) = C + 2x + y
C.
log (2x + y + 1) + x + 2y = C
D.
log (2x + y - 1) + x + 2y = C

Solution:

QUESTION: 20

The solution of the differential equation y(x²y + e^x)dx- e^xdy = 0 is

A.
3xy³ + e^x = cx
B.
xy³ + 3e^x = cx
C.
3x³y³ + e^x = cx
D.
x³y + 3e^x = 3yc

Solution:

Correct Answer :- D

Explanation : y(x²y + e^x)dx - e^xdy = 0

Dividing it by y²

(x² + e^x/y)dx - e^x/y² dy = 0

x² dx + (ye^x - e^x dy)/y² = 0

d(x³/3) + d(e^x/y) = 0

∫d(x³/3) + ∫d(e^x/y) = 0

=> (x³/3) + (e^x/y) = c

=> x³y + 3e^x = 3yc

Differential Equations - 12

20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Differential Equations - 12

The solution of the differential equation yy' + y² - x = 0, where c is a constant, is

Let k be a real constant. The solution of the differential equations satisfies the relation

If general solution of the differential equation ay",+ by"- cy’+ dy = 0 is linearly spanned by e^x sin x and cos x, then which one of the following holds?

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