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

Properties

  1. Associative
  2. Commutative
  3. Distributive over addition
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