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QUESTION: 1

The differential equation of all circles of radius a is given by

Solution:

**I Proof:** The family of circles of radius a is given by

(x - h)^{2} + (y - k)^{2} = a^{2}, .......(i)

where h and k are the parameters to be eliminated. Since there are two arbitrary constants, therefore equation (i) is to be differentiated twice and the order of the differential equation must be 2.

Now differentiating (i) twice, we get

and

Eliminating k and h from equations (i), (ii) and (iii), the required differential equation of the circles of radius a is given by

QUESTION: 2

A differential equation is said to be linear if

Solution:

**Definition :** A differential equation is said to be linear if.

i. the dependent variable and each of its derivatives occurs to the first degree only and

ii. no products of dependent variable and or its derivatives occur.

QUESTION: 3

Which of the following differential equations is linear?

Solution:

**Remark :** Observe that the dependent variable y and its derivatives

occur to the first degree and there are no products of them in statement (d).

The product of y and dy/dx occurs in (a), (b) and (c)

QUESTION: 4

Which of the following differential equations is linear but not of first degree

Solution:

The differential equations in (a), (b) and (c) are of first degree but none of them is linear.

QUESTION: 5

A differential equation of first degree

Solution:

(i)(a) is not correct because a differential equation of first degree may not be linear. For example, the differential equation.

is of first degree but non-linear,

(ii) (b) is not correct because a differential equation of first, degree is not necessarily of first order. The differential equation (i) is of first, degree but of second order.

(iii) (d) is not correct because a differential equation of first degree may be of first order and may not be linear.

QUESTION: 6

A linear differential equation

Solution:

Any differential equation which is not of first degree can not be linear.

QUESTION: 7

Which one of the following is a standard form of the first order differential equation of first degree?

Solution:

The differential equation

is of first order and first degree. However, it may or may not be linear.

M(y, y)dx + N(x, y)dy = 0

can be rewritten as

and therefore is of first degree and first order. It also may or may not be linear.

QUESTION: 8

If M and N are functions of x and y, then the equation Mdx + Ndy = 0 is exact if

Solution:

The necessary and sufficient condition for the differential equation

M(x, y)dx + N(x, y)dy = 0 to be exact is that

Proof: Condition

**Remark :** An exact differential equation can be written as du = 0. where u is some function of x and y.

QUESTION: 9

The necessary and sufficient condition for the differential equation M(x, y)dx + N(x, y)dy = 0 to be exact is that

Solution:

QUESTION: 10

If P(x) and Q(y) are arbitrary functions of x and y respectively, then the differential equation P(x)dx + Q(y)dy = 0

Solution:

**Proof :** The given differential equation is

P(x)dx+Q(y)dy = 0 ...(i)

Comparing it with

Mdx + Ndy = 0 ...(ii)

∴ M = P(x) and N = Q(y)

is always satisfied.

⇒ D.E. (i) is always exact.

QUESTION: 11

If P(y) and Q(x) are arbitrary functions of y and a respectively, then the differential equation P(y)dx+ Q(x)dy = 0

Solution:

Proof : The given differential equation is

P(y)dx+Q(x)dy = 0 ...(i)

Comparing (i) with

Mdx + Ndy= 0 ...(ii)

M = P(y) and N = Q(x)

Differential equation (i) will be exact if and only if

...(iii)

QUESTION: 12

Which of the following differential equation is exact?

Solution:

Consider the differential equation in (a),

(y^{2} + x) dx + (y^{3} + x)dy = 0 ... (i)

so that DE (i) is not exact.

(ii) The differential equation in (b) can be rewritten as

(x + y - 1) dx - (x + y + 1 ) = 0 .....(ii)

so that DE (ii) is not exact

(iii) The differential equation in (c) is

2xy dx + (x + y^{2}) dy = 0 .....(iii)

so that DE (iii) is not exact.

Thus none of the differential equations in (a), (b) and (c) is exact.

QUESTION: 13

Which of the following differential equations is exact?

Solution:

The differential equation in (a) is exact.

**Remark : **Verify yourself that (a) is exact but (b), (c) or fd) are not exact.

QUESTION: 14

Which of the following provides a solution of the exact differential equation

+ {x + log x - x sin y}dy = 0

Solution:

The given differential equation can be written as

QUESTION: 15

Which of the following provides a solution of the exact differential equation

Solution:

The given differential equation can be rewritten as

QUESTION: 16

Which of the following is not an exact differential?

Solution:

∴ expressions in (a), (b) and (d; are exact.

Since the expression

xdy - ydx

in (c) can not be expressed as du for some function u of x and y, therefore the expression in (c) is not exact.

QUESTION: 17

Which of the following is an exact differential?

Solution:

QUESTION: 18

Which of the following is an exact differential?

Solution:

Since the expression in (a) can be written as an exact expression

QUESTION: 19

Which of the following is not an exact differential?

Solution:

The expression in (a) is not exact. Hence (a) is the correct answer.

**Remark :** Note that expression in (a)

Comments about Integrating Factor

**Definition :** Suppose that tne differential equation. M dx + N dy = 0 ...(i)

is not exact but the differential equation

is exact, where μ = F(x, y) for a suitably chosen function F. Then μ is called an integrating factor of the differential equation (i).

**Ex. : **The differential equation

(3ty + 4xy^{2}) dx + (2x + 3x^{2}y) dy = 0 ...(ii)

is not exact. But if we choose

μ = x^{2} y.

then the differential equation

becomes exact.

∴ μ = x^{2} y is an integrating factor of (ii).

Rules for finding an Integrating Factor :

**Rule : 1:** If Mx ± Ny ≠ 0. and is homogeneous in x and y then the integrating factor is

**Ex. : **Consider the differential equation

(x^{2}y - 2xy^{2})dx- (x^{3} - 3x^{2}y)dy = 0 ...(iii)

verify that

(i) equation (iii) is noi exact and (ii)

Intergrating factor is given by

Hence the equation

should be exact. Now equation (iv) can be written as

**Rule : 2. If
**

is a function of x alone, say f(x), then the integrating factor μ is given by

is a function of y alone, say φ(y), then the integrating factor μ is given by

QUESTION: 20

The integrating factor of the differential equation xdx - ydx = xy^{2}dx is

Solution:

Noto that

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