# Ordinary Differential Equations:Cheat Sheet/Few Useful Definitions

## Wronskian of Two Functions

### Definition

Wronskian of two functions, $y_{1},y_{2}$ is given by $W_{y_{1},y_{2}}(x)=\left|{\begin{matrix}y_{1}&&y_{2}\\y_{1}'&&y_{2}'\end{matrix}}\right|$ ### Useful Facts

• If two functions $y_{1},y_{2}$ are linearly dependent in an interval, then it's Wronskian vanishes in that interval.

## Laplace Transforms

### Definition

${\mathcal {L}}\{f(t)\}=F(s)=\int _{0}^{\infty }e^{-st}f(t)dt$ ### Properties

1. ${\mathcal {L}}\{af+bg\}=a{\mathcal {L}}\{f\}+b{\mathcal {L}}\{g\}\,,$ 2. ${\mathcal {L}}\{e^{at}f(t)\}(s)=F(s-a)\,$ for $s>\alpha +a$ .
3. If $F(s)={\mathcal {L}}\{f(t)\}$ , then ${\mathcal {L}}\{f'(t)\}=sF(s)-f(0)$ 4. Similarly, ${\mathcal {L}}\{f''(t)\}=s^{2}F(s)-sf(0)-f'(0)$ ### Laplace Transform of Few Simple Functions

1. ${\mathcal {L}}\{1\}={1 \over s}$ 2. ${\mathcal {L}}\{e^{at}\}={1 \over s-a}$ 3. ${\mathcal {L}}\{\cos \omega t\}={s \over s^{2}+\omega ^{2}}$ 4. ${\mathcal {L}}\{\sin \omega t\}={\omega \over s^{2}+\omega ^{2}}$ 5. ${\mathcal {L}}\{1\}={1 \over s}$ 6. ${\mathcal {L}}\{t^{n}\}={n! \over s^{n+1}}$ ## Convolution

### Definition

$f(t)*g(t)=\int _{0}^{t}f(u)g(t-u)dt$ 1. Associative
2. Commutative