Differential Equations - 2

# Differential Equations - 2

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## 20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Differential Equations - 2

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Differential Equations - 2 - Question 1

### The solution of the differential equation

Detailed Solution for Differential Equations - 2 - Question 1

we have (D3 - 9D)y = cos x
implies D(D2 - 9)y = cos x

So, complete solution is

Differential Equations - 2 - Question 2

### Consider the differential equation y" + 6y' + 25y - 0 with initial condition y(0) = 0. Then, the general solution of the IVP is

Detailed Solution for Differential Equations - 2 - Question 2

we have (D2 + 6D + 25)y = 0

So, y =e-3x [A cos 4x + B sin 4x]
but y(0) = 0 implies A = 0
Hence, y = Be-3x sin 4x

Differential Equations - 2 - Question 3

### The solution of the differential equation y" + 4y = 0 subject to y(0) = 1, y' (0) = 2 is

Detailed Solution for Differential Equations - 2 - Question 3

we have (D2 + 4) = 0
So, solutions is y = c1 sin 2x + c2 cos 2x
but y(0)= 1 implies c2 = 1
and y'(0) = 2 implies 2c1
= 2 implies c1 = 1
Hence y = sin 2x + cos 2x

Differential Equations - 2 - Question 4

General solution of the differential equation  is given by

Detailed Solution for Differential Equations - 2 - Question 4

we have xdy = (y + xe-y/x)dx
implies xdy - ydx = xe-y/x dx

Differential Equations - 2 - Question 5

The Wronskian of the function x |x| is zero for

Differential Equations - 2 - Question 6

The general solution of y" - m2y = 0 is

Detailed Solution for Differential Equations - 2 - Question 6

Explanation : y''- m2y = 0

r2 - m2 = 0

r = +-m

y = Aemx + Be-mx

Let A=C+D and B=D-C

= Cemx + Demx - Ce-mx + De-mx

= C(emx-e-mx) + D(emx+e-mx)

= Csinh(mx) + Dcosh(mx)

Differential Equations - 2 - Question 7

The orthogonal trajectories of the curves y2 = 3x3+x + c are

Detailed Solution for Differential Equations - 2 - Question 7

Differentiating w.r.t. x, we get

Replacing  the equation of orthogonal trajectory is given by

implies
Integrating both sides, we get

implies 2 tan-1 3x + 3 In |y| = k

Differential Equations - 2 - Question 8

The general solution of the differential equation

Detailed Solution for Differential Equations - 2 - Question 8

is satisfied, So, equation is exacct
...(i)
...(ii)

Adding (i) and (ii), writing the common term once.

Differential Equations - 2 - Question 9

The solution of the initial value problem xy' - y = 0 with y(1) = 1 is

Detailed Solution for Differential Equations - 2 - Question 9

Differential Equations - 2 - Question 10

If y'1 (x) = 3y1(x) + 4y2(x) and y'2 (x) = 4y1(x) + 3y2(x), then y1(x) is

Detailed Solution for Differential Equations - 2 - Question 10

we have (D - 3)y1 - 4y2 = 0
4y2 + (3 - D)y1 = 0
Solving equations, we get

So,

Differential Equations - 2 - Question 11

If ex + xy + x sin y + ey = c is the general solution of an exact differential equation, then the differential equation is

Detailed Solution for Differential Equations - 2 - Question 11

Taking total differential

Differential Equations - 2 - Question 12

The solution of the differential equation

Detailed Solution for Differential Equations - 2 - Question 12

we have

and
Since,
So, differential equation is exact
Now,

and

So, solution of differential equation

Differential Equations - 2 - Question 13

The differential equation (2x2 + by2)dx + cxydy = 0 is made exact by multiplying the integrating factor 1/x2. Then the relation between and c is

Detailed Solution for Differential Equations - 2 - Question 13

Multiplying the differential equation by 1/x2, we get
It is exact
So,
implies
2b + c = 0

Differential Equations - 2 - Question 14

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

Detailed Solution for Differential Equations - 2 - Question 14

Differential Equations - 2 - Question 15

If y = In (sin (x + a)) + b, where a and b are constants, is the primitive, then the corresponding lowest order differential equation is

Detailed Solution for Differential Equations - 2 - Question 15

we have y = In sin (x + a ) + b
implies y'= cot (x + a)
implies y" = - cosec2 (x + a)

Differential Equations - 2 - Question 16

The solution of the initial value problem

Detailed Solution for Differential Equations - 2 - Question 16

Differential Equations - 2 - Question 17

What is the degree of the non-homogeneous partial differential equation,

Detailed Solution for Differential Equations - 2 - Question 17

Degree of an equation is defined as the power of the highest derivative present in the equation. Hence from the equation, the degree is 5.

Differential Equations - 2 - Question 18

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

Detailed Solution for Differential Equations - 2 - Question 18

Differential Equations - 2 - Question 19

Consider the differential equation ( x + y + 1) dx + (2x + 2y + 1) dy = 0. Which of the following statements is true?

Differential Equations - 2 - Question 20

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

Detailed Solution for Differential Equations - 2 - Question 20

(x - h)2 + (y - k)2 = a2 ...(i)
where h, k are parameters
Differentiating w.r.t. x, we get
( x - h ) + ( y - k ) y' = 0 ...(ii)
Differentiating w.r.t. x again 1 + ( y - k)
y11 + y12  = 0
implies   (iii)
From (ii), we get ( x - h ) = - ( y - k)y'
...(iv)
Putting (x - h) and (y - k) in (i), we get

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