Differential Equations - 12


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

Differential Equations - 12 for Mathematics 2023 is part of Topic-wise Tests & Solved Examples for IIT JAM Mathematics preparation. The Differential Equations - 12 questions and answers have been prepared according to the Mathematics exam syllabus.The Differential Equations - 12 MCQs are made for Mathematics 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Differential Equations - 12 below.
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Differential Equations - 12 - Question 1

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

Detailed Solution for Differential Equations - 12 - Question 1


Differential Equations - 12 - Question 2

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

Detailed Solution for Differential Equations - 12 - Question 2


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

Differential Equations - 12 - Question 3

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

Detailed Solution for Differential Equations - 12 - Question 3

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

Differential Equations - 12 - Question 4

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

Detailed Solution for Differential Equations - 12 - Question 4

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

Differential Equations - 12 - Question 5

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

Detailed Solution for Differential Equations - 12 - Question 5



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


It ’s orthogonal trajectory will be D.E.

Differential Equations - 12 - Question 6

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

Detailed Solution for Differential Equations - 12 - Question 6

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

Differential Equations - 12 - Question 7

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

Detailed Solution for Differential Equations - 12 - Question 7




 

Differential Equations - 12 - 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.

Detailed Solution for Differential Equations - 12 - Question 8

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


Differential Equations - 12 - Question 9

Given that   Which one of the following is always true?

Detailed Solution for Differential Equations - 12 - Question 9

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

Differential Equations - 12 - 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

Detailed Solution for Differential Equations - 12 - Question 10



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

Intercept on Y-axis.

Intercept on X-axis:

Differential Equations - 12 - Question 11

 Solution of the equation 

Detailed Solution for Differential Equations - 12 - Question 11

0
 

Differential Equations - 12 - Question 12

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

Detailed Solution for Differential Equations - 12 - Question 12


Differential Equations - 12 - Question 13

The solution of the differential equation 

Detailed Solution for Differential Equations - 12 - Question 13


Differential Equations - 12 - Question 14

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

Detailed Solution for Differential Equations - 12 - Question 14

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 = cex-y

Differential Equations - 12 - Question 15

The solution of the equation 

Detailed Solution for Differential Equations - 12 - Question 15



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

Differential Equations - 12 - Question 16

The solution of the differential equation 

Detailed Solution for Differential Equations - 12 - Question 16



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

Differential Equations - 12 - Question 17

The solution of the equation 

Detailed Solution for Differential Equations - 12 - Question 17


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

Differential Equations - 12 - Question 18

The degree and order of differential equation  are respectively

Detailed Solution for Differential Equations - 12 - Question 18

Differential Equations - 12 - Question 19

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

Detailed Solution for Differential Equations - 12 - Question 19




Differential Equations - 12 - Question 20

The solution of the differential equation y(x2y + ex)dx- exdy = 0 is

Detailed Solution for Differential Equations - 12 - Question 20

Correct Answer :- D

Explanation : y(x2y + ex)dx - exdy = 0

Dividing it by y2

(x2 + ex/y)dx - ex/y2 dy = 0

x2 dx + (yex - ex dy)/y2 = 0

d(x3/3) + d(ex/y) = 0

∫d(x3/3) + ∫d(ex/y) = 0

=> (x3/3) + (ex/y) = c

=> x3y + 3ex = 3yc

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