Differential Equation

Basic Concepts and Ideas

Differential Equation: – An equation involving one dependent variable and its derivatives with respect to one or more independent variable is called as differential equation.
Example. \frac{dy}{dx}=-ky,
Ordinary differential equation: – An ordinary differential equation is one in which there is only one independent variable, so that all the derivatives occurring in it are ordinary derivatives.
Ex. \left(\frac{dy}{dx}\right)^2+2xy=e^{-x}
Partial differential equation: – A Partial differential equation is one involving more than one independent variable, so that all the derivatives occurring in it are partial derivatives.
Example. \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0
Order of differential equation: – The order of the differential equation is the order of the highest derivatives present.
Example. \frac{dy}{dx}=-ky (First order)
\left(\frac{dy}{dx}\right)^2+2xy=e^{-x} (First order)
x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+x^2y=0 (Second order)

Degree of differential equation: – The degree of differential equation is the exponent of highest order derivatives present in a differential equation after the differential equation has been rationalised and made free of fractional powers with regards to all derivatives present.
Ex. \frac{dy}{dx}=-ky(Degree 1)
\left(\frac{dy}{dx}\right)^2+2xy=e^{-x} (Degree 2)
x^2\frac{d^2y}{dx^2}+x\left(\frac{dy}{dx}\right)^3+xy=0 (Degree 1)
First order differential equation: -An equation contains \frac{dy}{dx} , y and function of x is called as first order differential equation.
Ex. \frac{dy}{dx}=f(x,y)
Solution of differential equation: – Any function free from derivatives, satisfying the differential equation identically is said to be a solution of differential equation.
Example. x+yy^\prime=0 has a solution x^2+y^2=1
Differentiating with respect to x
\Rightarrow\ x+yy^\prime=0 which is a solution of x^2+y^2=1.

n^{th} order ordinary differential equation: –
An n^{th} order differential equation in general form is

(1)   \begin{equation*}F\left(x,y,\frac{dy}{dx},\frac{d^2y}{dx^2},\frac{d^3y}{dx^3},..,\frac{d^ny}{dx^n}\right)=0\end{equation*}

Solution of (1) containing arbitrary constant is called as general solution.
Any solution obtained from the general solution by giving a particular value to the arbitrary constants is known as particular solution.

Initial Value problem: – A differential equation together with an initial condition is known as initial value problem, which can be expressed in the form of: –
\frac{dy}{dx}=f(x,y), y(x_0)=y_0. It means at x =x_0, y = y_0

Singular solution: -A differential equation may sometimes have an additional solution that cannot be obtained from the general solution and is then called a singular solution.


Take the first order differential equation y^\prime=f(x,y) (1)
If Equation (1) can be reduced in the form of g(y)y^\prime=f(x) (i.e. if we can separate the variables in x and y) Then its solution of equation (1) is