Differential Equation

A differential equation relates a function $\psi$ and it’s derivatives. Below we list a few types of equations and their solutions. These are often needed to solve other problems in physics. Below we have a table of solutions:

2nd Order Differential Equation

$$\begin{array}{rll}0& =a\ddot{\psi }(t)+b\dot{\psi }(t)+c\psi (t)& \\ & \Downarrow & \text{Factorising ax2+bx+c=a(x−x1)(x−x2)}\\ \psi (t)& =\{\begin{array}{cc}A{e}^{x_{1}t}+B{e}^{x_{2}t},& x_1\ne x_2\\ Ax{e}^{x_{1}t}+B{e}^{x_{1}t},& x_1=x_2\end{array}& \end{array}$$

If $b=0$ above, we can map $a\mapsto \frac{a}{c}$ to get the simpler solution of $0=a\ddot{\psi }(t)+\psi (t)$, for $A,B\in \mathbb{C}$:

$$\psi (t)=\{\begin{array}{cc}A\cos (\sqrt{a}t)+B\sin (\sqrt{a}t),& a\ge 0\\ 0& a=0\\ A{e}^{\sqrt{a}t}+B{e}^{-\sqrt{a}t},& a\le 0\end{array}$$

Integrating factors

$$\begin{array}{rll}\dot{\psi }(x)+f(x)\psi (x)& =g(x)& \\ & \Downarrow & h(x):=\int f(x)dx\\ \psi (x)& =e^{-h(x)}\int g(x)e^{h(x)}dx& \end{array}$$