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This mock test of Differential Equations - 12 for Mathematics helps you for every Mathematics entrance exam.
This contains 20 Multiple Choice Questions for Mathematics Differential Equations - 12 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

The solution of the differential equation yy' + y^{2} - x = 0, where c is a constant, is

Solution:

QUESTION: 2

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

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?

Solution:

Linearly independent solutions of ay" + by”+cy' + dy = 0 are e^{x} cos x and sin x, so roots of aD^{3} + bD^{2} + 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 y_{1} = e^{x} and y_{2} = xe^{x}. Then a particular solution of y^{2} - 2y' + y + e^{x} sin x is

Solution:

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

QUESTION: 5

Orthogonal trajectories of the family of curves (x - 1)^{2} + y^{2} + 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

Solution:

QUESTION: 8

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

Solution:

Integrating factor of (2x^{2} + by^{2}) 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

Solution:

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

Intercept on Y-axis.

Intercept on X-axis:

QUESTION: 11

Solution of the equation

Solution:

0

QUESTION: 12

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

Solution:

QUESTION: 13

The solution of the differential equation

Solution:

QUESTION: 14

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

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

Solution:

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

QUESTION: 16

The solution of the differential equation

Solution:

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

whose solution is

QUESTION: 17

The solution of the equation

Solution:

.......(i)

which is linear differential equation, so solution is

QUESTION: 18

The degree and order of differential equation are respectively

Solution:

QUESTION: 19

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

Solution:

QUESTION: 20

The solution of the differential equation y(x^{2}y + e^{x})dx- e^{x}dy = 0 is

Solution:

**Correct Answer :- D**

**Explanation : **y(x^{2}y + e^{x})dx - e^{x}dy = 0

Dividing it by y^{2}

(x^{2} + e^{x}/y)dx - e^{x}/y^{2} dy = 0

x^{2} dx + (ye^{x} - e^{x} dy)/y^{2} = 0

d(x^{3}/3) + d(e^{x}/y) = 0

∫d(x^{3}/3) + ∫d(e^{x}/y) = 0

=> (x^{3}/3) + (e^{x}/y) = c

=> x^{3}y + 3e^{x} = 3yc

### Differential Equations

Doc | 5 Pages

- Differential Equations - 12
Test | 20 questions | 60 min

- Differential Equations - 18
Test | 20 questions | 60 min

- Differential Equations - 13
Test | 20 questions | 60 min

- Differential Equations - 14
Test | 20 questions | 60 min

- Differential Equations - 8
Test | 20 questions | 60 min