# Differential Equations

## First Order

A first order differential equation has as it's highest derivative. For the two methods below, it is important the equation is in the correct form specified.

### Seperating Variables

For an equation of the form

The solution is

### Integrating Factors

For an equation of the form

An integrating factor can be found such that:

Multiplying through by gives

Then, applying he product rule backwards gives a solution:

## Second Order

A second order ODE has the form:

The equation is homogeneous if .

The auxillary equation is

This gives two roots and , which determine the complementary function:

RootsComplementary Function
and both real
, both real
and

The complementary function is the solution. Sometimes, initial conditions will be given which allow the constants and to be found.

### Non-Homogeneous Systems

If the system is non-homogenous, ie , then a particular integral is needed too, and the solution will have form . The particular integral is found using a trial solution, then substituting it into the equation to find the coefficients. Note that if the particular integral takes the same form as the complementary function, an extra will need to be added to the particular integral for it to work, it would become

Trial Solution
const const
polynomial
or

### Example

Auxillary equation:

Complementary function is therefore:

System is non-homogeneous, so have to find a particular integral. For this equation , so the p.i. is .

Substituting this into the original equation:

Comparing coefficients:

The general solution is therefore:

Using initial conditions to find constants, for

For

Particular solution for given initial conditions is therefore: