Differential Equations Class 12 Notes Maths Chapter 9 - CBSE

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 = ax² + bx , y = x² − 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