# First Order

First Order Differential Equations with worked examples

**Contents**

- Examples With Separable Variables Differential Equations
- Linear Type Of Differential Equation
- Equations That Can Be Reduced To The Linear Form
- General Solution Of The Bernoulli Equation
- Homogeneous Equations
- The Method Of Solution For Homogeneous Equations
- The Exceptional Case Of Homogeneous Equations
- Exact Equations
- Page Comments

## Examples With Separable Variables Differential Equations

This article presents some working examples with separable differential equations.**Definition**Separable Differential Equations are differential equations which respect one of the following forms :

- where
*F*is a two variable function,also continuous. - , where
*f*and*g*are two real continuous functions.

### Rational Functions

- A
**rational**function is a real function respecting where are polynomials.

Example:

##### Example - Simple Differential Equation

Problem

Solve:

Workings

As the equation is of first order, integrate the function twice, i.e.
and

Solution

### Trigonometric Functions

- A
**trigonometric**function is a real function where contains one or more of the trigonometric functions :

Example:

##### Example - Simple Cosine

Problem

Workings

This is the same as
which we integrate in the normal way to yield

Solution

### Physics Examples

Example:

##### Example - Potential example

Problem

If
and at r=a and at r=b

*a*and*b*are the radii of concentric spherical conductors at potentials of respectively, then*V*is the potential at a distance*r*from the centre. Find the value of*V*if:\frac{d}{dr}\left(r^2 \frac{dV}{dr} \right)=0}

Workings

\therefore\;\;\;\;\;\;r^2\frac{dV}{dr} \right)=A

*V*and

*r*and Thus

Solution

## Linear Type Of Differential Equation

Equations of the type Where*P*and

*Q*are function of

*x*( but not of

*y*) are said to be

**linear**of the first order

Example:

##### Example - Rational equation

Problem

Workings

If each side of te equation is multiplied by x the equation becomes:-
i.e
Hence integrating
This equation has been solved by using the obvious integrating factor x. It is possible to find a
more general solution by using R as and integrating factor.
Consider the following equation :
By Inspection the left hand side of this equation must reduce to (Ry)
This gives
Thus
This gives the rule that to solve multiply both sides by an integrating
factor of:-

Solution

Hence the Method of solving this type of equation is :

- Reduce the equation into the form
- Multiply through by the Integrating Factor:-
- The equation becomes :-

## Equations That Can Be Reduced To The Linear Form

A

Analogous for

**linear form**in 2 variables is given by whereAnalogous for

*n*variables .Example:

##### Example - Simple equations

Problem

Consider the equation:

Workings

Divide through by

PuttingSolution

Hence
Therefore
Or
This example is a particular case of

**The Bernoulli Equation**## General Solution Of The Bernoulli Equation

**Bernoulli equations**have an important property :

- they are nonlinear differential equations with known exact solutions.

This section is presenting the Bernoulli Equation.

and are functions of

*x*This can be reduced to a linear form by putting Therefore The original equation can be re-written as:## Homogeneous Equations

Any equation which can be put into the form:A

**homogeneous**polynomial is a polynomial whose monomials with nonzero coefficients all have the same total degree. For example : is homogenous.
is said to be

**Homogeneous**. To test whether a function of*x*and*y*can be written in the form of the right hand side, substitute for . If the result is in the form , i.e. all the*x's*cancel, then the test is satisfied and the equation is homogeneous.Example:

##### Example - Testing a function is homogeneous

Problem

Is the follow function homogeneous:

Workings

Substitute for y=vx,
or
or
As all the

*x*have cancelled out, the test is satisfied.Solution

Function is homogeous

## The Method Of Solution For Homogeneous Equations

Substitute in both sides of the equation Note. If y is a function of*x*then so is

*v*

Thus the equatican be re-written as:
Re-writing and Separating the variables:
Integrating
But

Example:

##### Example - Homogenous

Problem

Workings

Rearranging
Putting y = vx
i.e.
Integrating
Therefore
Therefore

Solution

Substituting for v
Therefore

## The Exceptional Case Of Homogeneous Equations

If the straight lines are parallel there is no finite point of intersection and the method of solving such equations is illustrated by the following example.
Put Z = 3y - 4x and thus
The equation can now be written as:
Integrating
Replacing Z the solution to the differential equation is :

## Exact Equations

A form is said to be

**exact**in a region if there is a function such as .
The expression
is an exact differential.

Thus the equation giving that i.e. is called an exact Equation.

Example:

##### Example - Exact differential

Problem

Solve

Workings

This equation is not exact as it stands but if it is multiplied through by it becomes:

Solution

The solution