How to solve second order linear differential equation

A second order differential equation is an ordinary differential equation (ODE) that involves only the second derivatives of an unknown function with respect to a single independent variable.

Mathematically, second order ODE is represented as:

For example:

Solving second order ODE

Suppose we have the following ODE to solve:

Let:

Hence, our differential equation would become:

We have two solutions now, $r=3 \textnormal{ and } r=2.$ Substituting their values in $y:$

To get a more general solution, we need to combine their multiples as follows:

where $A \textnormal{ and } B$ are constants.

Verification

We have $y=Ae^{2x}+Be^{3x}$. To check our answer, we need to calculate the following values:

Put the above values in the equation we have:

The above equality holds. Hence our solution is correct.

Note: Learn about how to solve first order linear differential equations.