Hans Georg Schaathun
September 2016
$$ay'' + by' + cy = 0$$
Andre orden: \(y''\)
Homogen: \(y'', y', y\) -- ingen konstantledd
Konstante koeffisientar: \(a, b, c\)
$$ay'' + by' + cy = 0$$
$$y = e^{rt}$$
$$ar^2e^{rt} + bre^{rt} + ce^{rt} = 0$$
$$ar^2 + br + c = 0$$
$$r = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
$$r = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \quad b^2>4ac$$
$$ r_1 = \frac{-b + \sqrt{b^2-4ac}}{2a} \quad\quad r_2 = \frac{-b - \sqrt{b^2-4ac}}{2a} $$
$$y = e^{r_1t} \quad\quad y = e^{r_2t}$$
$$y = Ae^{r_1t} + Be^{r_2t}$$
$$r = \frac{-b}{2a} \quad\quad b^2=4ac$$
$$y = e^{rt}$$
$$y = te^{rt}$$
$$r = \frac{-b}{2a} \quad\quad b^2=4ac$$
$$y = e^{rt} \quad\quad y = te^{rt}$$
$$y = A\cdot e^{rt} + B\cdot te^{rt}$$
$$r = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \quad b^2<4ac$$
$$r = \frac{-b \pm i\sqrt{4ac-b^2}}{2a}$$
$$\Re(r) = k = \frac{-b}{2a} $$ $$\Im(r)=\omega=\frac{\sqrt{4ac-b^2}}{2a}$$
$$y_1^* = e^{(k-i\omega)t} \quad\quad y_2^* = e^{(k+i\omega)t}$$
$$y^* = Ae^{(kt-i\omega)t} + B e^{(kt +i\omega)t}$$
$$y^* = Ae^{(kt-i\omega)t} + B e^{(kt +i\omega)t}$$
$$y^* = e^{kt}\big[Ae^{-i\omega t} + B e^{i\omega t}\big]$$
$$e^{i\omega t} = \cos (\omega t) + i\sin(\omega t)$$
$$e^{-i\omega t} = \cos (\omega t) - i\sin(\omega t)$$
$$\frac12( e^{i\omega t} + e^{-i\omega t}) = \cos (\omega t)$$
$$\frac1{2i}( e^{i\omega t} - e^{-i\omega t}) = \sin (\omega t)$$
$$y_1 = e^{kt}\cos(\omega t) \quad\quad y_2 = e^{kt}\sin(\omega t) $$
$$y = A\cdot e^{kt}\cos(\omega t) + B\cdot e^{kt}\sin(\omega t) $$