# Ordinary Differential Equations:Cheat Sheet/Few Useful Definitions

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## Wronskian of Two Functions

### Definition

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

### Useful Facts

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

## Laplace Transforms

### Definition

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

### Properties

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

### Laplace Transform of Few Simple Functions

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

## Convolution

### Definition

${\displaystyle f(t)*g(t)=\int _{0}^{t}f(u)g(t-u)dt}$

### Properties

1. Associative
2. Commutative
3. Distributive over addition