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This mock test of Differential Equations - 6 for Mathematics helps you for every Mathematics entrance exam.
This contains 20 Multiple Choice Questions for Mathematics Differential Equations - 6 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

The differential equation representing the family of circles touching y - axis at the origin is

Solution:

Equation of circles touching y axis is given by

( x - a^{2}) + y^{2} = a^{2} where a is parameter implies x^{2 }+ y^{2} - 2ax = 0

Differentiating w.r.t. x, we get

2x + 2yy' - 2a = 0

implies a = x + yy'

So, differential equation will be

x^{2} +y^{2}- 2x(x + yy^{1}) = 0

implies

So, differential equation is non-linear and of first order

QUESTION: 2

The general solution of the differential equation

Solution:

we have

on integrating, we get

implies

implies

implies

implies

implies

where c_{1}' is constant.

QUESTION: 3

If e^{2x} and xe^{2x} are particular solutions of a second order homogeneous differential equation with constant coefficients, then the equation is

Solution:

QUESTION: 4

If y^{a} is an integrating factor of the differential equation 2xy dx - (3x^{2} - y^{2}) dy = 0, then the value of a is

Solution:

QUESTION: 5

The particular integral of the following differential equation

Solution:

QUESTION: 6

If k is a constant such that satisfies the differential equation then k is equal to

Solution:

QUESTION: 7

The solution y(x) of the differential equations satisfying the conditions y(0) = 4,

Solution:

we have (D^{2} + 4D + 4)y = 0

implies D + 2)^{2} y = 0

So, solution is

but y(0) = 4 implies c_{1} = 4

So,

So, solution is given by (16x + 4) e^{-2x}

QUESTION: 8

The general solution of yy" - (y')2 = 0 is

Solution:

we have yy" - y'2 = 0

implies

on integrating, we get

implies In y' = In y + c

implies y' = c_{1}y

Again on integrating,

implies In y = c_{1}x + k

implies

QUESTION: 9

The solution of the differential equation

Solution:

we have (y^{2} sin x + x cos y ) dx - ( x sin y - 2y sin x) dy = 0

So, differential equation is exact

So, solution is given as

QUESTION: 10

A particular integral of the differential equation

Solution:

QUESTION: 11

Let y_{1}(x) and y_{2}(x) be twice differentiable functions on a interval I satisfying the differential equations Then y_{1}(x) is

Solution:

we have

implies ...(i)

implies ...(ii)

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

...(iii)

Adding (ii) and (iii), we get

and

So, complete solution is given as

QUESTION: 12

The general solution of the differential equation is

Solution:

Thus, solution is

implies

QUESTION: 13

The solution of the differential equation satisfying y(0) = 0 and dy/dx (0) = 3/2 is

Solution:

Here, C.F. = c_{1} sinh x + c_{2} cosh x

So,

but y(0) = 0 implies c_{2} = 0,

Thus

implies

At

So,

QUESTION: 14

An integrating factor of the differential equation 2xy dx + (y^{2} - x^{2}) dy = 0 is

Solution:

Here I.F. = 1/y^{2}, we get

Multiplying differential equation by 1/y^{2}

Now,

Since,

So, equation becomes exact

QUESTION: 15

If y = x cos x is a solution of an n^{th} order linear differential equation with real constant coefficients, then the least possible value of n is

Solution:

So, n = 4

QUESTION: 16

Let W[y_{1}(x), y_{2}(x)] is the Wronskian formed for the solutions y_{1}(x) and y_{2}(x) of the differential equation y" + a_{1}y' + a_{2}y = 0. If W ≠ 0 for some x = x_{0} in [a, b] then

Solution:

QUESTION: 17

The general solution of y' (x + y^{2}) = y is

Solution:

Here, y'(x +y^{2}) = y

implies

implies

So,

Thus, solution is

implies

implies x = cy + y^{2}

QUESTION: 18

The general solution of y' - 2^{x-y }is

Solution:

Here y' = 2^{x-y}

implies 2^{y} dy = 2^{x} dx

implies

implies 2^{x}-2^{y} = c'

QUESTION: 19

Solution of the differential equation xy' + sin 2y - x^{3} sin^{2 }y is

Solution:

implies

Let cot y= 2

Thus,

implies

So,

Hence, solution is

implies z = cot y = -x^{3} + cx^{2}

QUESTION: 20

A particular solution of the differential equation

(D^{4} + 2D^{2} - 3)y = e^{x} is

Solution:

### Differential Equations

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