Differential equation is any equation that involves derivatives or differential coefficient or differentials. In mathematics differential equation shows the relationships between physical quantities which appear in mathematical form.

One very common differential equation is Newton’s second law of Motion.
If an object of mass m moving with acceleration a is acted upon with a force F then Newton’s second law says:

$F = ma$
Acceleration can be written as a first derivative of velocity

F = $\frac{dv}{dt}$ or as a second derivative of displacement F = $\frac{d^2 s}{d^2 t}$

m$\frac{dv}{dt}$ = F(t, v)…………(1)

m$\frac{d^2 u}{dt^2}$ = F(t, u, $\frac{du}{dt}$)………(2)

Order of differential equation is the largest derivative present in the given equation. In the differential equation shown above (1) [with $\frac{dv}{dt}$] is first order differential equation., (2) is second order equation (with $\frac{d^2 s}{d^2 t}$).

Types of Differential Equations

Differential equations are ordinary and partial type.
In ordinary differential equation all the derivatives coefficient are differentiated with respect to single variable.
Example:

a y" + by' +cy = g(t)
ln (y)$\frac{d^2 y}{dx^2}$ = (3-y)$\frac{dy}{dx}$ + $y^3$

In partial differential equation has partial derivatives.
P$\frac{dy}{dx}$ = Q is said to be a homogeneous differential equation if P and Q are homogeneous functions of x and y with same degree.

Whereas non homogeneous differential equation means in given function the degree of X and Y are not same. Some common examples of non homogeneous function are exponential function, trigonometric function, high power of X, product of exponential and higher order of X functions, product of exponential and trigonometric function, and product of higher order of X and trigonometric functions.

If $\frac{dy}{dx}$ = f($\frac{y}{x}$)

To check if the equation is homogeneous substitute y = vx. If we get the result in the form of ƒ (v) which means all the x ‘cancelled then the equation is said to be homogeneous.

Example: Test if the given equation is homogeneous.

$\frac{dy}{dx}$ = $\frac{y^2 - x^2}{xy}$ .........................(1)

Substitute y = xv

=> dy = xdv + vdx

(1) => $\frac{dy}{dx}$ = $\frac{(xv)^2 - x^2}{x *(xv)}$

=
$\frac{x^2(v^2 - 1)}{x^2 v}$

= $\frac{v^2 - 1}{v}$

=> $\frac{dy}{dx}$ = $\frac{v^2 - 1}{v}$

$\rightarrow$ $\frac{xdv + vdx}{dx}$ = $\frac{v^2 - 1}{v}$ = v - $\frac{1}{v}$

=> xdv + vdx = vdx - $\frac{1}{v}$ dx

$\rightarrow$ dx + xvdv = 0

$\rightarrow$ vdv - $\frac{-1}{x}$ dx

$\rightarrow$ $\frac{v^2}{2}$ = -logx + logc

$\frac{1}{2}$$v^2$ = log|$\frac{c}{x}$|

To get the result in the form of x and y, again substitute v = $\frac{y}{x}$

=> $\frac{y^2}{2x^2}$ = log|$\frac{c}{x}$|

Difference Quotient

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