SHORT NOTES: Differential Equations

Introduction

Differential Equation involves independent variable,dependent variable,derivatives of dependent variable.

Example : $\frac{dy}{dx}+y=xe^x$

Here y is the dependent variable,x independent variable,(dy/dx) is the derivative of dependent variable y.

Differential equation(DE)

Ordinary DE Partial DE
(contain one independent variable) (contain more than one independent variable) $\dfrac{\partial }{{\partial x}}$ stands for the partial derivative

The order of a DE is the order of the highest derivative that occurs in the equation.

The degree of a DE is the degree of the highest order derivative that occurs in the equation, when all the derivatives in the equation are made of free from radicals and fraction(can be written as a polynomial in differential coefficient)

Order always defined but not degree. e.g. $\frac{d^2y}{dx^2}+\ sin (\frac{dy}{dx})=0$

Example : $\frac{d^3y}{dx^3}+\left (\frac{dy}{dx} \right )^3=xe^x$ Order = 3, degree = 1

Formation of ordinary differential equations

Differentiate the given equation w.r.t independent variable(say x) as many times as the number of arbitrary constants in it such that all the constants are eliminated forming the required differential equation.

Example : $y=ae^x$
$\frac{dy}{dx}=ae^x\Rightarrow \frac{dy}{dx}=y$

A differential equation is said to be non linear if

its degree is more than one
any differential coefficient has exponent more than one
exponent of dependent variable is more than one
product containing dependent variable and its diff. coefficient present

The form of Linear differential equation is

$p_0\left (\frac{d^ny}{dx^n} \right )+p_1\left (\frac{d^\left (n-1 \right )y}{dx^\left (n-1 \right )} \right )+...+p_{n-1}\left (\frac{dy}{dx} \right )=Q$

where $p_0,p_1,...,p_{n-1},Q$ are either constants or functions of independent variable x.

If Q=0 , then it is called homogenous linear differential equation (HLDE) .

Separation of variables method

The equation $\frac{dy}{dx}=f(x,y)$ is said to be in variable separable form if we can express it in the form $f(x)dx=g(y)dy$ .

Then we can get the solution by integrating both sides as

$\int f(x)dx=\int g(y)dy+c$

Example: $\frac{dy}{dx}=\frac{2y}{x}$

$\frac{dx}{x}=\frac{dy}{2y}$

$\int \frac{dx}{x}=\int \frac{dy}{2y}+c$

$\Rightarrow \log y=2\log x+\log c$

$\Rightarrow y=cx^2$

Linear first order differential equations:

Form $\frac{dy}{dx}+Py=Q$

where P and Q are either constants or functions of x.

Integrating Factor = $e^{\int P dx}$

Solution $ye^{\int P dx}=\int {Qe^{\int P dx}}dx+c$

Example:   $\frac{dy}{dx}+y=5$
I.F= $e^{\int dx}=e^x$
Soln :    $ye^x=\int {5e^x}dx+c$
$\Rightarrow ye^x=5e^x+c$
$\Rightarrow y=5+ce^{-x}$

Reducible to Linear Form(above)

Form $\frac{dy}{dx}+Py=Qy^n$
$\Rightarrow \frac{1}{y^{-n}}\frac{dy}{dx}+Py^{1-n}=Q$

Take $z=y^{1-n}$ and derive to replace y with z

Example:

   $\frac{dy}{dx}+4xy=xy^3$
$\Rightarrow \frac{1}{y^3}\frac{dy}{dx}+4xy^{-2}=x$
   $z=y^{-2}\Rightarrow -\frac{1}{2}\frac{dz}{dx}=\frac{1}{y^3}\frac{dy}{dx}$
$\frac{dz}{dx}-8xz=-2x$ (reduced form)
Solving it as above prob we get
$z{e^{-4x^2}}=\frac{e^{-4x^2}}{4}+c$
$\Rightarrow \frac{{e^{-4x^2}}}{y^2}=\frac{e^{-4x^2}}{4}+c$

ELECTRICAL ENGINEERING
ENGINEERING MATHEMATICS	Electric Circuits and Fields
Signals and Systems	Electrical Machines
Power Systems	Control Systems
Electrical and Electronic Measurements	Power Electronics and Drives
Analog Electronics	Digital Electronics

Differential Equations

No comments:

Post a Comment