# Ordinary Differential Equations:Cheat Sheet/Second Order Homogeneous Ordinary Differential Equations

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## With Constant Coefficients[edit | edit source]

### General Form[edit | edit source]

or , where

- is called the polynomial differential operator with constant coefficients.

### Solution[edit | edit source]

- Solve the auxiliary equation, , to get
- If are
**Real and distinct**, then**Real and equal**, then**Imaginary**, , then

## Euler-Cauchy Equations[edit | edit source]

### General Form[edit | edit source]

or where

- is called the polynomial differential operator.

### Solution[edit | edit source]

Solving is equivalent to solving

## General Homogenous ODE with Variable Coefficients[edit | edit source]

### If one particular solution is known[edit | edit source]

If one solution of a homogeneous linear second order equation is known, , original equation can be converted to a linear first order equation using substitutions and subsequent replacement .

#### Abel's identity[edit | edit source]

For the homogeneous linear ODE , Wronskian of its two solutions is given by

##### Solution with Abel's identity[edit | edit source]

Given a homogenous linear ODE and a solution of ODE, , find Wronskian using Abel’s identity and by definition of Wronskian, equate and solve for .

##### Few Useful Notes[edit | edit source]

- If are linearly dependent,
- If , for some , then .