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

## With Constant Coefficients

### General Form

${\displaystyle ay''+by'+cy=0}$ or ${\displaystyle p(D)y=0}$, where

${\displaystyle p(D)=aD^{2}+bD+c}$ is called the polynomial differential operator with constant coefficients.

### Solution

1. Solve the auxiliary equation, ${\displaystyle p(m)=0}$, to get ${\displaystyle m=\lambda _{1},\lambda _{2}}$
2. If ${\displaystyle \lambda _{1},\lambda _{2}}$ are
1. Real and distinct, then ${\displaystyle y(x)=Ae^{\lambda _{1}x}+Be^{\lambda _{2}x}}$
2. Real and equal, then ${\displaystyle y(x)=(Ax+B)e^{\lambda _{1}x}}$
3. Imaginary, ${\displaystyle \lambda _{i}=a\pm bi}$, then ${\displaystyle y(x)=(A\cos {bx}+B\sin {bx})e^{ax}}$

## Euler-Cauchy Equations

### General Form

${\displaystyle ax^{2}y''+bxy'+cy=0}$ or ${\displaystyle p(D)y=0}$ where

${\displaystyle p(D)=ax^{2}D^{2}+bxD+c}$ is called the polynomial differential operator.

### Solution

Solving ${\displaystyle ax^{2}y''+bxy'+cy=0}$ is equivalent to solving ${\displaystyle ay''+(b-a)y'+cy=0}$

## General Homogenous ODE with Variable Coefficients

### If one particular solution is known

If one solution of a homogeneous linear second order equation is known, ${\displaystyle y_{1}(x)\neq 0}$, original equation can be converted to a linear first order equation using substitutions ${\displaystyle y_{2}=y_{2}(x)z(x)}$ and subsequent replacement ${\displaystyle z^{'}(x)=u}$.

#### Abel's identity

For the homogeneous linear ODE ${\displaystyle y''+p(x)y'+q(x)y=0}$, Wronskian of its two solutions is given by ${\displaystyle W_{(}y_{1},y_{2})(x)=W(x_{0})e^{-\int _{x_{0}}^{x}p(x)dx}}$

##### Solution with Abel's identity

Given a homogenous linear ODE and a solution of ODE, ${\displaystyle y_{1}(x)}$, find Wronskian using Abel’s identity and by definition of Wronskian, equate and solve for ${\displaystyle y_{2}(x)}$.

##### Few Useful Notes
1. If ${\displaystyle y_{1},y_{2}}$ are linearly dependent, ${\displaystyle W(x)=0,\forall x}$
2. If ${\displaystyle W(x)=0}$, for some ${\displaystyle x}$, then ${\displaystyle W(x)=0,\forall x}$.