Math, asked by sanket12b, 3 months ago

The differential equations cos (Y-x) Ce^-x

Answers

Answered by shifarahman2008
2

Answer:

(i) An equation involving derivative (derivatives) of the dependent variable with

respect to independent variable (variables) is called a differential equation.

(ii) A differential equation involving derivatives of the dependent variable with

respect to only one independent variable is called an ordinary differential

equation and a differential equation involving derivatives with respect to more

than one independent variables is called a partial differential equation.

(iii) Order of a differential equation is the order of the highest order derivative

occurring in the differential equation.

(iv) Degree of a differential equation is defined if it is a polynomial equation in its

derivatives.

(v) Degree (when defined) of a differential equation is the highest power (positive

integer only) of the highest order derivative in it.

(vi) A relation between involved variables, which satisfy the given differential

equation is called its solution. The solution which contains as many arbitrary

constants as the order of the differential equation is called the general solution

and the solution free from arbitrary constants is called particular solution.

(vii) To form a differential equation from a given function, we differentiate the

function successively as many times as the number of arbitrary constants in the

given function and then eliminate the arbitrary constants.

(viii) The order of a differential equation representing a family of curves is same as

the number of arbitrary constants present in the equation corresponding to the

family of curves.

(ix) ‘Variable separable method’ is used to solve such an equation in which variables

can be separated completely, i.e., terms containing x should remain with dx and

terms containing y should remain with dy.

(x) A function F (x, y) is said to be a homogeneous function of degree n if

F (λx, λy )= λn

F (x, y) for some non-zero constant λ.

(xi) A differential equation which can be expressed in the form dy

dx = F (x, y) or

dx

dy = G (x, y), where F (x, y) and G (x, y) are homogeneous functions of degree

zero, is called a homogeneous differential equation.

(xii) To solve a homogeneous differential equation of the type dy

dx = F (x, y), we make

substitution y = vx and to solve a homogeneous differential equation of the type

dx

dy = G (x, y), we make substitution x = vy.

(xiii) A differential equation of the form dy

dx + Py = Q, where P and Q are constants or

functions of x only is known as a first order linear differential equation. Solution

of such a differential equation is given by y (I.F.) = ( ) Q I.F. × dx ∫

+ C, where

I.F. (Integrating Factor) = Pdx

e

.

(xiv) Another form of first order linear differential equation is

dx

dy + P1x = Q1

, where

P1

and Q1

are constants or functions of y only. Solution of such a differential

equation is given by x (I.F.) = ( ) Q × I.F. 1 dy ∫

+ C, where I.F. = P1dy

e

. 9.2 Solved Examples

Short Answer (S.A.)

Example 1 Find the differential equation of the family of curves y = Ae

2x

+ B.e

–2x

. Solution y = Ae

2x

+ B.e

dy

dx = 2Ae

2x

– 2 B.e

–2x

and

2

2

d y

dx

= 4Ae

2x

+ 4Be

–2x

Thus

2

2

d y

dx

= 4y i.e.,

2

2

d y

dx

– 4y = 0.

Example 2 Find the general solution of the differential equation dy

dx = y

x

.

Solution dy

dx = y

x

dy

y

= dx

x

dy

y

∫ = dx

x

⇒ logy = logx + logc ⇒ y = cx

Example 3 Given that dy

dx = yex

and x = 0, y = e. Find the value of y when x = 1.

Solution dy

dx = yex

dy

y

∫ = x e dx ∫

⇒ logy = e

x

+ c

Substituting x = 0 and y = e,we get loge = e

0+ c, i.e., c = 0 (loge = 1)

Therefore, log y = ex

. Now, substituting x = 1 in the above, we get log y = e ⇒ y = e

e

. Example 4 Solve the differential equation dy

dx +

y

x

= x

2.

Solution The equation is of the type + P = Q dy

y dx

, which is a linear differential

equation.

Now I.F. =

1

dx

x

∫ = e

logx

= x.

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