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QUESTION: 1

If f_{1} and f_{2} be the integrating factors of the differential equations respectively,

Solution:

The given differential equations are

......(i)

QUESTION: 2

Which of the following statements is correct?

Solution:

The infinite number of differential equations given by

for infinitely many different choices of Q(x), possess the same integrating factor

QUESTION: 3

The integrating factor of the differential equation is given by

Solution:

The given ciifferential equation is

The required Integrating factor is given by

[log x = t]

= e ^{log t} = t = log x

QUESTION: 4

The integrating factor of the differential equation is calculated as

Solution:

The given differential equation is

The required Integrating factor is given by

QUESTION: 5

The differential equation of the type is known a

Solution:

QUESTION: 6

Which of the following transformations reduces the differential equation sec y into the form by the substitution

Solution:

The given differential equation can be rewritten as

This implies that

∴ differential equation (i) reduces to

**Remark :** Observe that the transformation (ii) is suggested by form (i) of given differential eouation.

QUESTION: 7

Which of the following transformations reduces the differential equation into the form by the substitution

Solution:

Rewrite the given differential equation as

This suggests that the required transformation is

QUESTION: 8

The transformation v = 1/y transforms which the following differential equations into the form

Solution:

QUESTION: 9

The general solution of the differential equation where φ is a function of x alone, is given by

Solution:

The given differential equation is

.....(i)

Its Integrating Factor is given by

becomes exact.

This implies

QUESTION: 10

The equation xdy = ydx represents the family of

Solution:

The differential equation is x dy = y dx

This represents the family of straight lines passing through the origin.

QUESTION: 11

The equation xdy + ydx = 0 represents a family of

Solution:

The given differential equation can be written as

⇒ log y + log x = log c

⇒ yx = c

This represents the family of hyperbolas.

QUESTION: 12

The one parameter family y = or can be written as

Solution:

The given one parameter family is

y = cx^{2} .....(i)

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

....(ii)

On eliminating (c) from (i) and (ii). we get

QUESTION: 13

A curve is called an isogonal trajectoiy or an x trajectory of family f(x, y, c) = 0 if

Solution:

Statement (c) gives the correct definition of an isogonal trajectory.

QUESTION: 14

In order to find a family of orthogonal trajectories of the family of curves which of the following replacements is made

Solution:

Given family of the curves is

.....(i)

Its slope is dy/dx. If m is the slope of the orihogonal trajectories of the family (i) then we should have

QUESTION: 15

The orthogonal trajectories of the given family of curves y = cx^{2} are given by

Solution:

Then given family of curves is

y = cx^{k} ....(i)

.v —

Let us first find the differential equation satisfied by the family (i). For this we differentiate (i) w.r. t.

∴ The differential equation of the orthogonal trajectories will be obtained on replacing

⇒ Orthogonal trajectories are given by

QUESTION: 16

Which one of the following is a general solution of the differential equation xp^{2} + (y - x)p - y = 0 where

Solution:

The given differential equation

QUESTION: 17

Which one of the following is a general solution of the differential equation p^{2} - 2p cos hx + 1 = 0?

Solution:

**Hint:** Write

and do as in the previous question.

QUESTION: 18

Which one of the following is a general solution of the differential equation y = 2px + p^{2}y ?

Solution:

Some comments before the solution.

1. If the given equation in x, y and p can be solved for y, then we shall have

y = f(x,p) ...(i)

Differentiate (i) w.r. t. x. This will lead to a differentia! equation in p and x. Solve it for p and then eliminate p from the resulting equation and (i).

2. If the given equation in x, y and p can be solved for x, then we shail have

x = f(y, p) .......(ii)

Differentiate (ii), w.r. t. y and follow the same procedure as in case - i.

Now solution of the problem 82.

The given differential equation is

y = 2 px + p^{2}y ...... (iii)

... (iv)

which is solved for x, i.e. it expresses x as a function of y and p.

Now differentiating (iv) w.r. t. y, we get

Now eliminating p from equations (iii) and (v), we get

or y^{2} = 2cx + c^{2}

QUESTION: 19

Which one of the following is a general solution of the differential equation y - 2px = f(xp^{2})?

Solution:

The given differential equation is

Now eliminating p from equations (i) and (iii).

QUESTION: 20

The differential equation of the form y = x F(p) + f(p) is known as

Solution:

### Differential Equations

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