Differential Equation -

FIRST ORDER 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. $\frac{dy}{dx}=-ky,$
$\left(\frac{dy}{dx}\right)^2+2xy=e^{-x}$ ,
$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+x^2y=0$
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 $x^2+y^2=1$
$x^2+y^2=1$
Differentiating with respect to $x$
$\Rightarrow2x+2yy^\prime=0$
$\Rightarrow\ x+yy^\prime=0$ 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.

1.3 SEPARABLE DIFFERENTIAL EQUATION

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