Mathematics SOLUTION OF DIFFRENTIAL EQUATIONS BY THE METHOD OF SEPARATION OF VARIABLES

SOLUTION OF DIFFRENTIAL EQUATIONS BY THE METHOD OF SEPARATION OF VARIABLES

Solution Of a differential equation :

`text( T-1. Variables Separable :) `

If the differential equation can be expressed as; `f(x)dx + g(y)dy = 0` then this is said to be variable -separable type.

A general solution of this is given by `int f(x) dx + int g(y) dy =c`; where c is the arbitrary constant.

`text( T-2. Differential Equation Reducible to the Separable Variable Type:)`

`dy /dx = f (ax +by +c), a,b ne 0`

To solve this , substitute `t = ax+ by + c`. Then the equation reduces to separable type in the
variable `t` and `x` which can be solved.

Equations Reducible to the Homogenous Form :

Equation of the form `dy/dx = (ax+by +c) /(Ax +By +C) ` (`aB ne Ab` and `A+b ne 0`) can be reduced to a homogeneous
form by changing the variable `x,y`, to `X, Y` by writing `x = X +h` and `y = Y + k`; where `h, k` are constant
to be chosen so as to make the given equation homogeneous. We have

`dy /dx = (d (Y+k))/(d (X+h)) =(dY)/(dX)`

Hence the given equation becomes,

`dY/dX = (aX+ bY+ (ah+bk+c) )/(Ah + Bk + (Ah+Bk+C))`

Let `h` and `k` be chosen to satisfy the relation `ah + bk + c = 0` and `Ah + Bk + C = 0`.

Polar Coordinates :

Sometimes transformation to the polar co- ordinates facilitates separation of variables. In this
connection it is convenient to remember the following differentials. If `x = r cos theta`; `y= r sin theta` where `r` and `theta` both are variable.

(a) (i) `x dx + y dy = r dr` (ii) `x dy - y dx = r^2d theta`

Proof: `x = r cos theta` ; `y = r sin theta` `=> x^2 + y^2 = r^2`
`=> x dx + y dy = rdr`

Also `tan theta = y/x` `=> xdy - y dx = x^2 sec^2 theta d theta`
`=> xdy - ydx = r^2 d theta`

(b) If `x = r sec theta` & `y = r tan theta` then

(i) `x dx - ydy = r dr ` and (ii) `x dy - y dx = r^2 sec theta d theta`.

Proof: `x = r sec theta` and `y = r tan theta`

`=> x^2 - y^2 = r^2 => xdx - ydy = rdr`

` y/x = sin theta => xdy - ydx = x^2 cos theta d theta = r^2 sec theta d theta`