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This mock test of Test: Differential Equations & Multiple Integrals- 2 for GATE helps you for every GATE entrance exam.
This contains 30 Multiple Choice Questions for GATE Test: Differential Equations & Multiple Integrals- 2 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

Solution:

QUESTION: 2

The volume of an object expressed in spherical co-ordinates is given by

The value of the integral is

Solution:

QUESTION: 3

By a change of variable x (u, y) = uv, y (u, v) = v/u is double integral, the integrand f(x, y) changes to f(uv, v/u) φ (u,v). Then, φ (u, v) is

Solution:

QUESTION: 4

Consider the shaded triangular region P shown in the figure. What is

Solution:

The equation of the line AB is

QUESTION: 5

A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of (x + y)^{2} on path AB traversed in a counterclockwise sense is

Solution:

QUESTION: 6

The value of

Solution:

QUESTION: 7

A parabolic cable is held between two supports at the same level. The horizontal span between the supports is L. The sag at the mid-span is h. The equation of the parabola is where x is the horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is

Solution:

We know length of the curve f(x) between x = a and x = b given by

QUESTION: 8

A surface S(x,y)=2x+5y-3 is integrated once over a path consisting of the points that satisfy ( x +1)2+ (y − 1)2 = √2 . The integral evaluates to

Solution:

QUESTION: 9

The value of integral

Solution:

QUESTION: 10

The order of the differential equation

Solution:

The order of a differential equation is the order of the highest derivative involving in

equation, so answer is 2.

QUESTION: 11

The solution of the differential equation under the boundary conditions

(i) y =y_{1} At x = 0 and

(ii) y =y_{2} At x = ∞,

Where k, y_{1} and y_{2} are constants, is

Solution:

QUESTION: 12

A function n(x) satisfies the differential equation where L is a constant. The boundary conditions are: n(0)=K and n(∞)= 0. The solution to this equation is

Solution:

For finite solution c_{1} = 0

QUESTION: 13

The solution of the differential equation

Solution:

Given differential equation is

Integra ling we get

QUESTION: 14

If then what is y(e)?

Solution:

It is linear differential equation.

QUESTION: 15

The solution of dy/dx = y^{2} with initial value y (0) = 1 bounded in the interval

Solution:

QUESTION: 16

The solution to the differential equation f’’(x)+4f’(x)+4f(x)=0 is

Solution:

Let y(x) = emx (m ≠ 0)be the trial sol^{n} .Auxiliary equation. m^{2} + 4m+ 4 = 0 ⇒(m+ 2)^{2} = 0

In particular, when A =1,B =1,then f(x) = (1 + x)e^{−2x}

= e^{−2x} + xe^{−2x}

QUESTION: 17

The above equation is a

Solution:

Since, the differential equation cannot be expressed in x/y or y/x form, therefore, it is an example of non-homogeneous differential equation.

QUESTION: 18

Which of the following is a solution of the differential equation

Solution:

Here p = 4andq = 3.The given equation becomes

QUESTION: 19

The Blasius equation,

Solution:

f is non linear.

QUESTION: 20

Given that .. .x+ 3x= 0, and x(0)=1, x(0) = 0 what is x(1)?

Solution:

Auxiliary equn of x^{11} + 3x = 0 is

m^{2} + 3 = 0

QUESTION: 21

The degree of the differential equation

Solution:

QUESTION: 22

The solution for the differential equation with the condition that y = 1 at x = 0 is

Solution:

QUESTION: 23

A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in

Solution:

By the given condition

QUESTION: 24

Solution of the differential equation represents a family of

Solution:

QUESTION: 25

The general solution of

Solution:

Let y = e^{mx} (m ≠ 0) be the trial solution.

QUESTION: 26

A body originally at 60ºC cools down to 40ºC in 15 minutes when kept in air at a temperature of 25ºC. What will be the temperature of the body at the end of 30 minutes?

Solution:

QUESTION: 27

The partial differential equation that can be formed from z = ax + by + ab has the form

Solution:

QUESTION: 28

With K as constant, the possible solution for the first order differential equation is

Solution:

QUESTION: 29

The boundary-value problem y^{n} + λy = 0, y(0) = y(λ) = 0 will have non-zero solutions if and only if the values of λ are

Solution:

QUESTION: 30

The solution of the differential equation with the condition that y = 1 at x = 1, is

Solution:

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