9. Differential Equations

Def.: equation with both relevant function and its derivates.

Linearity: differential equation of shape $a_n(x)y^{(n)}(x)+a_{n-1}(x)y^{(n-1)}(x)+\cdots +a_1(x)y^{\prime }(x)+a_0(x)y(x)=s(x)$

Order: max derivative order

Störfunktion (perturbation): $s(x)$. If $s(x)=0$: homogenous, else inhomogenous.

Linear, Homogenous

Super Position Principle

Theorem

$y_1(x),y_2(x)=$ solutions of a linear, homogeneous differential equation $⟹y(x)=C_1{y}_1(x)+C_2{y}_2(x),C_1,C_2\in \mathbb{R}$ also a solution.

Proof:

• ${\left[C_1{y}_1(x)+C_2{y}_2(x)\right]}^{(i)}$ = $C_1{y}_1(x{)}^{(i)}+C_2{y}_2(x{)}^{(i)}$
• Split $a_i(x)y^{(i)}(x)$ using above, separate using commutativity, pull out $C_1,C_2$
• Since both are zero by def., their sum is zero $⟹$ the homogenous differential equation is solved

Fundamental Solutions

Theorem

A linear, homogeneous differential equation of order $n$ has $n$ solutions $y_1(x),\dots ,y_n(x)$ such that the general solution is given by

$$y(x)=C_1{y}_1(x)+\cdots +C_n{y}_n(x),C_1,\dots ,C_n\in \mathbb{R}$$

These solutions are called fundamental solutions.

Proof:

• Order $n$ $⟹$ $n$ derivatives $⟹$ $n$ integrations
• Each integration introduces a free constant $+C_i$ $⟹$ $n$ free variables

Solution Approach

For linear, homogenous equations: $y(x)={\mathrm{e}}^{\lambda x}$ always solution

$$\begin{array}{rl}& {\mathrm{e}}^{\lambda x}(a_n{\lambda }^n+a_{n-1}{\lambda }^{n-1}+\dots +a_0)\\ & =a_n{\lambda }^n+a_{n-1}{\lambda }^{n-1}+\dots +a_0\\ & =0\end{array}$$

Setup of Differential Equations