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QUESTION: 1

If I_{1} and I_{2} be the integrating factors of the differential equations (1 + xy)ydx + (1 - xy)xdy = 0 and (x^{2}y - 2xy^{2})dx - (x^{3} - 3x^{2}y)dy = 0 respectively, then

Solution:

QUESTION: 2

The integrating factor of differential equation (xy sin xy + cos xy)ydx + (xy sin xy - cos xy)xdy = 0 is

Solution:

QUESTION: 3

The integrating factor of the differential equation (x^{3} + y^{2} + 2x)dx + 2ydy = 0 is

Solution:

**Hint : **Here

QUESTION: 4

The integrating factor of the differential equation (y^{4} + 2y)dx + (xy^{3} + 2y^{4} - 4x)dy = 0 is

**Note:** A differential equation is said to be with separated variables if it can be written in the form of P(x)dx + Q(y)dy = 0

Solution:

**Hint : **

is a function of y alone.

QUESTION: 5

Which of the following differential equations is with separated variables?

Solution:

Differential equation in (a) can be written as e^{x}e^{y}dx + sin x cos y dy = 0

QUESTION: 6

Which of the following differential equations is with separated variables?

Solution:

All the three differential equations in statements (a), (b) and (e) are with separated variables.

Do it yourself.

QUESTION: 7

The differential equation (x + y) [dx - dy] = dx + dy which is not with separated variables, can be transformed into one which is with separated variables, by the substitution

Solution:

**Proof: **Let x + y mv

QUESTION: 8

The differential equation which is not with separated variables, can be transformed into one which is with separated variables, by the substitution.

Solution:

**Proof:** Under the transformation

the given differential equation reduces to

Hence the variables are separated.

QUESTION: 9

Which of the following differential equations is not homogeneous one?

Solution:

**Definition : **Homogeneous Differential Equation.

The differential equations which can be expressed as

are called homogeneous differential equations. Thus differential equation (a) is homogeneous because this can be written as

Differential equation (b) is homogeneous because this can be written as

which is of the type

P(u) dv - Q(x) dx = 0 ...(ii)

(i.e.seperated variables)

Hence it is exact.

QUESTION: 10

Which one of the following statements is correct?

Solution:

QUESTION: 11

The solution curves of the given differential equation xdx + ydy = 0 are given by a family of

Solution:

QUESTION: 12

The solution eurves of the given differential equation xdx - dy = 0 are given by a family of

Solution:

QUESTION: 13

A first order first degree homogeneous differential equation

Solution:

QUESTION: 14

A first order first degree homogeneous differential equation

Solution:

QUESTION: 15

Which of the following transformations reduces a homogeneous differential equation of first order and first degree into one with separated variables?

Solution:

QUESTION: 16

The transformation y = vx reduces the given homogeneous differential equation

Solution:

The tranformation is

∴ given differential equation reduces to

QUESTION: 17

Which of the following transformations reduces the given differential equation in to homogeneous one?

Solution:

The transformation

given in (b) reduces the differential equation

which is homogeneous.

QUESTION: 18

A general linear differential equation of first order can be written as

Solution:

The most general linear first order differential equation is given by

Remark : DE in (a) may not be linear and DE in (c) is not most general.

QUESTION: 19

Which of the following differential equations is linear, homogeneous and of first order?

Solution:

DE in (a) is linear, homogeneous and first order.

DE in (b) is linear, first order but non-homogeneous.

DE in (c) is of first order, non-linear & non-homogeneous DE in (d) is of first order, linear but non-homogeneous.

QUESTION: 20

The integrating factor of the differential equation depends upon

Solution:

The integrating factor is given by

Therefore the integrating factor depends upon P(x) only.

### Differential Equations

Doc | 5 Pages

### Lec 17 | MIT 18.03 Differential Equations, Spring 2006

Video | 45:44 min

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- Differential Equations - 8
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