Ordinary Differential Equations:Cheat Sheet/Few Useful Definitions

Wronskian of Two Functions

Definition

The 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 on an interval, then their Wronskian vanishes on that interval.

Laplace Transforms

Definition

The Laplace transform of $f$ at a complex number $s\in \mathbb {C}$ is

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

Properties

• Linearity: ${\mathcal {L}}\{af+bg\}=a{\mathcal {L}}\{f\}+b{\mathcal {L}}\{g\}\,$

If $F(s)={\mathcal {L}}\{f\}(s)$ then:

• ${\mathcal {L}}\{e^{at}f(t)\}(s)=F(s-a)\,$ for $s>\alpha +a$
• ${\mathcal {L}}\{f'\}(s)=sF(s)-f(0)$
• ${\mathcal {L}}\{f''\}(s)=s^{2}F(s)-sf(0)-f'(0)$

Laplace Transform of Few Simple Functions

• ${\mathcal {L}}\{1\}={1 \over s}$
• ${\mathcal {L}}\{e^{at}\}={1 \over s-a}$
• ${\mathcal {L}}\{\cos \omega t\}={s \over s^{2}+\omega ^{2}}$
• ${\mathcal {L}}\{\sin \omega t\}={\omega \over s^{2}+\omega ^{2}}$
• ${\mathcal {L}}\{1\}={1 \over s}$
• ${\mathcal {L}}\{t^{n}\}={n! \over s^{n+1}}$

Convolution

Definition

The convolution of $f$ and $g$ is

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

Properties

Convolution is:

1. Associative
2. Commutative
3. Distributive over addition