Ordinary Differential Equations:Cheat Sheet/Few Useful Definitions

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.

Definition
The Laplace transform of $$f$$ at a complex number $$s \in \Complex$$ 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 $$s > \alpha + a$$
 * $$\mathcal{L}\{f'\}(s) = sF(s) - f(0)$$
 * $$\mathcal{L}\{f''\}(s) = s^2F(s) - sf(0) - f'(0)$$

Laplace Transform of Few Simple Functions

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

Definition
The convolution of $$f$$ and $$g$$ is

$$f(t)*g(t)=\int_0^t f(u)g(t-u)dt$$

Properties
Convolution is:
 * Associative
 * Commutative
 * Distributive over addition