# Differential equations/Linear homogeneous differential equations

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## Second Order Differential Equations

A second order differential equation has a given function and terms including the first and second derivatives of that function. There are many techniques that are employed to find solutions to these problems. Different types of problems need different techniques to solve them.

## Homogeneous

### Definition

For second order differential equations, a homogeneous equation is one where the right hand side is zero (e.g. ${\displaystyle y''+by'+cy=0}$  ).

### Solution

1. A general solution to this equation is ${\displaystyle y=e^{mx}}$  . Substitute this into the equation.
2. After dividing by the exponential part, the equation should resolve to the form ${\displaystyle m^{2}+bm+c=0}$  .
3. Solve the quadratic equation using the quadratic formula to get the roots of ${\displaystyle m}$ .
4. If there are 2 roots, the solution is ${\displaystyle y=Ae^{m_{1}x}+Be^{m_{2}x}}$  . If there is 1 root, the solution is ${\displaystyle y=Ae^{mx}+Bxe^{mx}}$  .