Ordinary Differential Equations:Cheat Sheet/Few Useful Definitions

The Wronskian of two functions ${\displaystyle y_{1},y_{2}}$ is given by

${\displaystyle W_{y_{1},y_{2}}(x)=\left|{\begin{matrix}y_{1}&&y_{2}\\y_{1}'&&y_{2}'\end{matrix}}\right|}$

• If two functions ${\displaystyle y_{1},y_{2}}$ are linearly dependent on an interval, then their Wronskian vanishes on that interval.

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

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

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

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

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

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

The convolution of ${\displaystyle f}$ and ${\displaystyle g}$ is

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

Convolution is:

1. Associative
2. Commutative
3. Distributive over addition