3609 Inhomogeneous Case Variation of Parameters Formula

The fundamental matrix $\Phi(t)$ also provides a very compact and efficient integral formula for a particular solution to the inhomogeneous equation $\boldsymbol{x'} = A(t)\boldsymbol{x} + F(t)$. (presupposing of course that one can solve the homogeneous equation $\boldsymbol{x'} = A(t)\boldsymbol{x}$ first to get $\Phi$.) In this short note we give the formula (with proof!) and one example.

Variation of parameters: (solving inhomegeneous systems)
(H) $\boldsymbol{x'}=A(t)\boldsymbol{x}\rightsquigarrow \Phi(t) =$ fundamental matrix
(I) $\boldsymbol{x'}=A(t)\boldsymbol{x}+F(x)$
Variation of parameters formula for solution to (I) (just like order 1 DE's): $$\boldsymbol{x}=\Phi\cdot(\int\Phi^{-1}\cdot Fdt+C)$$ proof
General homogeneous solution: $\boldsymbol{x}=\Phi\cdot \boldsymbol{c}$ for a constant vector $\boldsymbol{c}$
Make $c$ variable $\rightsquigarrow$ trial solution $\boldsymbol{x}=\Phi\cdot\boldsymbol{v}(t)$
Plug this into (I): $\boldsymbol{x'}=A\boldsymbol{x}+F\rArr \Phi'\cdot\boldsymbol{v}+\Phi\cdot\boldsymbol{v'}=A\Phi\cdot\boldsymbol{v}+F$
Now substitute for $\Phi'=A\Phi$ $$\begin{aligned} \rArr &A\Phi\cdot\boldsymbol{v}+\Phi\cdot\boldsymbol{v'}=A\Phi\cdot\boldsymbol{v}+F\\ \rArr &\Phi\cdot\boldsymbol{v'}=F\\ \rArr &\boldsymbol{v'}=\Phi^{-1}\cdot F\\ \rArr &\boldsymbol{v}=\int\Phi^{-1}\cdot F+C\\ \rArr &\boldsymbol{x}=\Phi\cdot(\int\Phi^{-1}\cdot Fdt+C) \end{aligned}$$ Definite integral version of variation of parameters
$$\boldsymbol{x}(t)=\Phi(t)(\int_{t_0}^{t}\Phi^{-1}(u)\cdot F(u)du+C), \text{ where } C=\Phi^{-1}(t_0)\cdot\boldsymbol{x}(t_0)$$

Example. Solve $$\boldsymbol{x'}=\begin{pmatrix} 6&5\\1&2 \end{pmatrix}\boldsymbol{x}+\begin{pmatrix} e^t\\e^{5t} \end{pmatrix}$$ Notation: $A = <script type="math/tex; mode=display">\begin{pmatrix}6&5\\1&2\end{pmatrix}$ , $F=<script type="math/tex; mode=display">\begin{pmatrix}e^t\\e^{5t}\end{pmatrix}$
Fundamental matrix $$\Phi=\begin{pmatrix} e^t&5e^{7t}\\-e^t&e^{7t} \end{pmatrix}$$ $$\Phi^{-1}=\frac{e^{-8t}}{6}\begin{pmatrix} e^{7t}&-5e^{7t}\\e^t&e^t \end{pmatrix}$$ Variation of parameters: $$\begin{aligned} \boldsymbol{x}&=\Phi\int \Phi^{-1}\cdot Fdt\\ &=\Phi \int \frac{e^{-8t}}{6}\begin{pmatrix} e^{7t}&-5e^{7t}\\e^t&e^t \end{pmatrix} \cdot \begin{pmatrix} e^t\\e^{5t} \end{pmatrix}dt\\ &=\Phi \int \frac{1}{6} \begin{pmatrix} 1-5e^{4t}\\e^{-6t}+e^{-2t} \end{pmatrix}dt\\ &=\frac{1}{6}\Phi\begin{pmatrix} t-\frac{5}{4}e^{4t}+c_1\\ -\frac{1}{6}e^{-6t}-\frac{1}{2}e^{-2t}+c_2 \end{pmatrix}\\ &=\frac{1}{6}\begin{pmatrix} te^t-\frac{5}{4}e^{5t}+c_1e^t-\frac{5}{6}e^{t}-\frac{5}{2}e^{5t}+5c_2e^{7t}\\ -e^tt+\frac{5}{4}e^{5t}-c_1e^t-\frac{1}{6}e^{t}-\frac{1}{2}e^{5t}+c_2e^{7t} \end{pmatrix}\\ &=\frac{1}{6}[te^t\begin{pmatrix} 1\\-1 \end{pmatrix}+e^{5t}\begin{pmatrix} -15/4\\3/4 \end{pmatrix}+e^t\begin{pmatrix} -5/6\\-1/6 \end{pmatrix}+c_1e^t\begin{pmatrix} 1\\-1 \end{pmatrix}+c_2e^{7t}\begin{pmatrix} 5\\1 \end{pmatrix}] \end{aligned}$$