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Differensiallikninger (førsteordens (seperabel (\(g\left(y\right)\cdot…
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
\(r=p\pm q\sqrt{-1}\)
\(y=e^{px}\left(C\cdot\cos\left(qx\right)+D\cdot\sin\left(qx\right)\right)\)