Differensiallikninger

førsteordens

$y'+f\left(x\right)\cdot y=g\left(x\right)$

$y\cdot e^{\int f\left(x\right)dx}=\int\left(g\left(x\right)\cdot e^{\int f\left(x\right)dx}\right)dx$

$y'+\frac{k}{x}y=g\left(x\right)$

$y\cdot e^{k\ln\left(x\right)}=\int\left(g\left(x\right)\cdot e^{k\ln\left(x\right)}\right)dx$

seperabel

$g\left(y\right)\cdot y'=f\left(x\right)$

$y'=\frac{dx}{dy}$

andreordens

$ay''+by'+cy=0$

reell

to løsninger

$y=C\cdot e^{r_{1}x}+D\cdot e^{r_{2}x}$

en løsning

$y=e^{r_{1}x}\left(C+D\cdot x\right)$

$r=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

imaginær

$r=p\pm q\sqrt{-1}$

$y=e^{px}\left(C\cdot\cos\left(qx\right)+D\cdot\sin\left(qx\right)\right)$