Formulas for the solution of second order ODE. Here is the collection of some standard formulas for the solution of second order ODE.

### Formulas for the solution of second order ODE

General form of a second-order linear ODE

The general form of any second order linear ordinary differential equation is

Here and are functions of .

Any second-order ordinary differential equation that can not be written in this form, is a non-linear one.

A second-order linear ODE is of two types. One is a homogeneous second-order ordinary differential equation.

The other one is a non-homogeneous second-order ordinary differential equation.

##### Homogeneous second-order ordinary differential equation

The general form of any homogeneous second order ordinary differential equation is

(1)

Here both and are real numbers.

Now in order to solve any homogeneous second order ordinary differential equation, one has to go through a few steps.

First of all, I’ll determine the characteristic equation.

This characteristic equation has another name too. That is the auxiliary equation.

In this example, characteristic equation is

Here is the coefficient of . And, is the coefficient of .

Next job is to solve the characteristic equation to get the values of .

Now there are two main possibilities for the values of .

One is, the values of are real. The other one is that these values are complex numbers.

Among the real values of , there are two further options.

First one is that these real numbers are identical. The second option is that the numbers will be different.

So all together, there are three choices for the roots of the characteristic equation – real different roots, real equal roots and complex conjugate roots.

Let me take the first option. So the roots are real and different, say,

Then the solution of the first order homogeneous ODE (1) is

Here both and are constants.

Now I consider the second option. So the roots are real and equal, say,

Then the solution of the first order homogeneous ODE (1) is

Here both and are constants.

At the end comes the third option. Here the roots are complex conjugate numbers, say,

Then the solution of the first order homogeneous ODE (1) is

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