Differential Equations - 6


20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Differential Equations - 6


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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. The solved questions answers in this Differential Equations - 6 quiz give you a good mix of easy questions and tough questions. Mathematics students definitely take this Differential Equations - 6 exercise for a better result in the exam. You can find other Differential Equations - 6 extra questions, long questions & short questions for Mathematics on EduRev as well by searching above.
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 - a2) + y2 = a2 where a is parameter implies x2 + y2 - 2ax = 0
Differentiating w.r.t. x, we get 
2x + 2yy' - 2a = 0 
implies a = x + yy'
So, differential equation will be 
x2 +y2- 2x(x + yy1) = 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 c1' is constant.

QUESTION: 3

If e2x and xe2x are particular solutions of a second order homogeneous differential equation with constant coefficients, then the equation is

Solution:
QUESTION: 4

If ya is an integrating factor of the differential equation 2xy dx - (3x2 - y2) 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 (D2 + 4D + 4)y = 0
implies D + 2)2 y = 0
So, solution is 
but y(0) = 4 implies c1 = 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' = c1y
Again on integrating, 
implies In y = c1x + k
implies 

QUESTION: 9

The solution of the differential equation

Solution:

we have (y2 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 y1(x) and y2(x) be twice differentiable functions on a interval I satisfying the differential equations  Then y1(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. = c1 sinh x + c2 cosh x


So, 
but y(0) = 0 implies c2 = 0,
Thus 
implies 
At 

So, 

QUESTION: 14

An integrating factor of the differential equation 2xy dx + (y2 - x2) dy = 0 is

Solution:

Here I.F. = 1/y2, we get
Multiplying differential equation by 1/y2

Now, 
Since, 
So, equation becomes exact 

QUESTION: 15

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

Solution:


So, n = 4

QUESTION: 16

Let W[y1(x), y2(x)] is the Wronskian formed for the solutions y1(x) and y2(x) of the differential equation y" + a1y' + a2y = 0. If W ≠ 0 for some x = x0 in [a, b] then 

Solution:
QUESTION: 17

The general solution of y' (x + y2) = y is

Solution:

Here, y'(x +y2) = y
implies 
implies 
So, 
Thus, solution is 
implies 
implies x = cy + y2

QUESTION: 18

The general solution of y' - 2x-y is

Solution:

Here y' = 2x-y
implies 2y dy = 2x dx
implies 
implies 2x-2y = c'

QUESTION: 19

Solution of the differential equation xy' + sin 2y - x3 siny is 

Solution:


implies 
Let cot y= 2
Thus, 
implies 
So, 
Hence, solution is 
implies z = cot y = -x3 + cx2

QUESTION: 20

A particular solution of the differential equation
(D4 + 2D2 - 3)y = ex is

Solution:

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