## Chapter:9

# What are Differential Equations ?

## Differential Equation

A differential equation is an equation involving independent variable, dependent variable and derivative of dependent variable(s) w.r.t. independent variable(s).

**Notation of differential equation:**

$$\frac{\text{dy}}{\text{dx}}=\text{y}_1\space \text{or}\space (\text{y}^{,}), \frac{d^2y}{dx^2}=\text{y}_2 \text\space {\text{or}} (\text{y}^{,,}),………\frac{d^ny}{dx^n}=\text{y}_n$$

**Order Of Differential Equation**

It is the order of the highest derivative which occurs.

**Degree Of Differential Equation**

It is the power of the highest order derivative.

## General And Particular Solutions Of A Differential Equation

**General Solution**

The solution which contains arbitrary constants is called a general solution (primitive) of the differential equation. For example :

y = ax^{2} + bx , y = x^{2} − C

**Particular Solution**

The solution obtained from the general solution by giving particular values to the arbitrary constants is a particular solution of the differential equation.,

## Solution Of Differential Equations By Method Of Separation Of Variables

If any differential equation, it is possible to express all the functions of x and dx on one side and all the functions of y and dy on the other side, the variables are said to be separable.

**Rule**

$$\text{To solve the equation}\space \frac{dy}{dx}=\text{h(y). g(x),} \text{where g is the function of x only and h is the function of y only.}\\ \space \text{• Given equation is} \space \frac{dy}{dx} =\text{h(y). g(x)}\\ \space\text{• Separating the variables , i.e.,}\frac{dy}{h(y)}=\text{g(x).dx}\\ \text{• Integrating both sides and adding an arbitrary constant on one side.}\\\text{i.e.,} \int\frac{1}{\text{h(y)}}\text{dy}=\int \text{g(x) . dx}\\ \text{Thus, the solutions of given differential equation is the form: H(y)= G(x)+C}\\ \text{Here , H(y) and G(x) are the anti derivatives of}\space\frac{1}{\text{h(y)}} \text{and g(x) respectively and C is the arbitrary constant.}$$

**Note**

Addition of an arbitrary constant is must on one side of the differential equation.

## Solutions Of Homogeneous Differential Equations Of First Order And First Degree Of The Type : dy/dx=f(y/x)

A differential equation of the form dy/dx=f1(x,y)/f2(x,y) where f1(x,y) and f2(x,y) are both homogeneous functions of the same degree in x and y, i.e., an equation of the form dy/dx=f(y/x) is called a homogeneous differential equation.

**Rule**

To solve a homogeneous differential equation of the type dy/dx= F (x,y) = f(y/x)

- Given equation is dy/dx= f(y/x)
- Put y = v . x in the given equation.
- Differentiating equation (y = v.x) with respect to x ,we get dy/dx= v +x (dv/dx)
- Substituting the value of dy/dx in given equation, we get x dv/dx= f(v) -v
- Separating the variables dv/f (v) - v=dx/x

$$\bull\space \text{Integrating both sides}\int\frac{\text{dv}}{\text{f(v)-v}}=\int\frac{1}{x}\space \text{dx}$$

- Above equation gives general solution of the differential equation when we replace v by y/x

## Solutions Of The Linear Differential Equation Of The Type

dy/dx+ py = q, where p and q are functions of x or constants.

dx/dy+ py = q, where p and q are functions of y or constants.

**Linear Differential Equation**

An equation of the form dy/dx+ py = q, where p and q are functions of x alone or constant, is called a linear differential equation of first order in y.

**Working rule to solve the linear differential equation**

- Write the given equation in the form dy/dx+ py = q
- Find the integrating factor (I.F.) = e
^{ ∫ p.dx} - The solution of the differential equation is y [I.F]=∫q. (I.F.)dx + C.