Variation of parameters

Description

Variation der Konstanten ist ein Verfahren, mit dem sich lineare inhomogene DGLSs lösen lassen.

Variation of parameters for order-one linear ODEs

Examine the following order-one Linear ODE $x^{\prime }(t)=a(t)x(t)+b(t),x(t_0)=x_0$ We will first calculate the solution of the homogenous part and then do a variation to determine a solution to the nonhomogeneous part. Separation of variables tells us that the solution to $x^{\prime }(t)=a(t)x(t)$ is given by: ${\lambda }_h(t)=x_0{e}^{A(t)},A(t):={\int }_{t_0}^{t}a(s)ds$ We will vary ${\lambda }_h$ by a factor $c(t)$ to obtain the solution to the nonhomogenous differential equation. Since we have $\lambda (t_0)={\lambda }_h(t_0)=x_0$ we also need $c(t_0)=1$: $\lambda (t)=c(t){\lambda }_h(t)$ This will be our ansatz. Note that this is always possible since ${\lambda }_h(t)$ is an exponential and therefore strictly positive. I don’t get why but the next step just works. If we now plug the ansatz back into the nonhomogenous differential equation and simplify, we get $c^{\prime }(t)x_0{e}^{A(t)}=b(t)\text{or}{x}_0{c}^{\prime }(t)=b(t)e^{-A(t)}$ Integrate both side and obtain (also remember that $c(t_0)=1$): $x_{0}c(t)=x_0+{\int }_{t_0}^{t}b(s)e^{-A(s)}ds$ Plugging back in into $\lambda (t)=c(t){\lambda }_h(t)$, we get $\lambda =x_0{e}^{A(t)}+\underset{t_0}{\overset{t}{\int }}b(s)e^{-A(s)}ds\cdot e^{A(t)}$ Or in terms of the homogenous solution $\lambda ={\lambda }_h(t)+{\lambda }_h(t)\underset{t_0}{\overset{t}{\int }}b(s){\lambda }_h(s{)}^{-1}ds\cdot$ The solution consists of one summand being the solution to the homogenous problem and then a second more complicated summand, dependent on $b(t)$.

I wonder, if there is some more geometric approach to this. I need to read Arnold’s again.

Variation of parameters for a system of ODEs

Betrachte ein Lineares Differentialgleichungssystem erster Ordnung $x^{\prime }=A(t)x+g(t)$ zum Initial value problem $x(\tau )=\xi$. Definiere die Übergangmatrix als $\Lambda (t,\tau )=\Phi (t)\Phi (\tau {)}^{-1}$. Sie gibt an, wie sich alle Lösungen über die Zeit von $\tau$ zu $t$ verändern. Das Anfangswertproblem hat nun die eindeutige maximale Lösung ${\lambda }_{(\tau ,\xi )}(t)=\Lambda (t,\tau )\xi +\Lambda (t,\tau )\underset{\tau }{\overset{t}{\int }}\Lambda (s,\tau {)}^{-1}g(s)ds$ wobei $\Phi (t)$ die Fundamentalmatrix des homogenen Teils von $(1)$ und $\Phi (t{)}^{-1}$ dessen Inverse zum Startzeitpunkt $\tau$ ist.

Variation of parameters for a system of ODEs with constant matrix

Ist $A(t)=A\in M_d(\mathbb{R})$ also konstant, dann ist die Fundamentalmatrix $\Phi (t)=e^{tA}$. Außerdem ist $\Phi (t)\Phi (\tau {)}^{-1}=e^{(t-\tau )A}$ die Übergangsmatrix für $x^{\prime }=Ax$ Damit vereinfacht sich die Lösungsformel: ${\lambda }_{(\tau ,\xi )}(t)=e^{(t-\tau )A}\xi +e^{(t-\tau )A}\underset{\tau }{\overset{t}{\int }}{e}^{-(s-\tau )A}g(s)ds$