**MATHS :: Lecture 07 :: Differential Equations**

**Differential Equations**

Differential equation is an equation in which differential coefficients occur.

A differential equation is of two types

(1) Ordinary differential equation

(2) Partial differential equation

**An ordinary differential equation** is one which contains a single independent variable.

Example:

**A partial differential equation** is one containing more than one independent variable.

**Examples**

Here we deal with only ordinary differential equations.

**Definitions**

**Order**

The order of a differential equation is the order of the highest order derivative appearing in it.

Order 1

Order -2

**Degree**

The degree of a differential equation is defined as the degree of highest ordered derivative occurring in it after removing the radical sign.

First-order linear ODEs3-Laplace transform

First-order linear ODEs2- constant coefficients

**Example**

Give the degree and order of the following differential equation.

1) 5 (x+y) + 3xy = x2 degree -1, order -1

2) - 6 + xy = 20 degree -3, order -2

3) = 3+1

Squaring on both sides

= 9+6+1

degree -1, order 2

4) =

1+3+3+=

degree – 2, order – 2

**Note**

If the degree of the differential equations is one. It is called a linear differential equation.

**Formation of differential equations**

**Given the solution of differential equation, we can form the corresponding differential equation. Suppose the solution contains one arbitrary constant then differentiate the solution once with respect to x and eliminating the arbitrary constant from the two equations. We get the required equation. Suppose the solution contains two arbitrary constant then differentiate the solution twice with respect to x and eliminating the arbitrary constant between the three equations.**

**Solution of differential equations**

- Variable separable method,
- Homogenous differential equation

iii) Linear differential equation

**Variable separable method**

Consider a differential equation = f(x)

Here we separate the variables in such a way that we take the terms containing variable x on one side and the terms containing variable y on the other side. Integrating we get the solution.

**Note **

The following formulae are useful in solving the differential equations

- d(xy) = xdy +ydx

**Homogenous differential equation**

** **Consider a differential equation of the form

(i)

where f1 and f2 are homogeneous functions of same degree in x and y.

Here put y = vx

(ii)

Substitute equation(ii) in equation (i) it reduces to a differential equation in the variables v and x. Separating the variables and integrating we can find the solution.

**Linear differential equation**

A linear differential equation of the first order is of the form , Where p and Q are functions of x only.

To solve this equation first we find the integrating factor given by

Integrating factor = I.F =

Then the solution is given by

where c is an arbitrary constant.

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