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

$W_{y_1,y_2}(x)=\left|\begin{array}{ccc}{y}_1& & y_2\\ {y_1}^{\prime }& & {y_2}^{\prime }\end{array}\right|$

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 $s\in ℂ$ is

$ℒ\{f\}(s)=F(s)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}f(t)dt$

Properties

• Linearity: $ℒ\{af+bg\}=aℒ\{f\}+bℒ\{g\}$

If $F(s)=ℒ\{f\}(s)$ then:

• $ℒ\{e^{at}f(t)\}(s)=F(s-a)$ for $s>\alpha +a$
• $ℒ\{f^{\prime }\}(s)=sF(s)-f(0)$
• $ℒ\{f^{″}\}(s)=s^{2}F(s)-sf(0)-f^{\prime }(0)$

Laplace Transform of Few Simple Functions

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

Convolution

Definition

The convolution of $f$ and $g$ is

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

Properties

Convolution is:

1. Associative
2. Commutative
3. Distributive over addition