ODE Cheat Sheet: Few Relevant Definitions

It’s been a while since I wrote here, and this is an attempt to resurrect the blog, and get inspiration to contribute something for Wikibooks as well.

This is a part of the book “Ordinary Differential Equations: Cheat Sheet” that I am compiling in Wikibooks, and it will be published in five instalments before/along with finalising the book in Wikibooks. And just like Examples and Counter Examples Series (Which again I am planning to compile in Wikibooks), this will also be edited as I learn.

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, $\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f''(0)$

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