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Andenordensdifferentialligninger
Begyndelsesværdier
$$y'' + y' - 2y=0, \s y(0),\s y'(0)=-1$$
$$r^{2}+r-2=0 \arrow = \frac{-1\pm\sqrt{1+8}}{2}=\begin{cases} r_1=1\ r_2=-2 \end{cases}$$
$$y=A \cdot e^{x}+B\cdot e^{-2x}$$
$$y'=A \cdot e^{x} -B \cdot 2 \cdot e^{-2x}$$
Indsætter værdier
$$ \begin{align} y(0)=A \cdot e^{0} + B \cdot e^{-2 \cdot 0} = 0 &\arrow A+B=2 \ y'=A \cdot e^{x} -B \cdot 2 \cdot e^{-2x} &\arrow A-2B=-1 \end{align} $$
Trækker de to ligninger fra hinanden
$$(A+B)-(A-2B)=2-(-1) \arrow A=1,\s B=1$$