I have the following IVP:
$$\ddot{x} + 2\dot{x} – 8x = 4$$ subject to the initial values $$x(0) = 0 \\ \dot{x}(0) = 0$$
I am asked to solve it using matrix method (I don't know if it is the correct translation to English, but basically, it wants me to solve this through linear algebra).
I don't have much experience in solving differential equations with linear algebra, but I know how to solve something like a system of equations involving $\frac{dx}{dt}$, $\frac{dy}{dt}$ and $\frac{dz}{dt}$ by using $\dot{X}=AX$ and etc.
Best Answer
You could model it like this: $$ \frac{d}{dt} \left[ \begin{matrix} x \\ \dot{x} \end{matrix} \right] + \left[ \begin{matrix} 0 & -1 \\ -8 & 2 \end{matrix} \right] \left[ \begin{matrix} x \\ \dot{x} \end{matrix} \right] = \left[ \begin{matrix} 0 \\ 4 \end{matrix} \right] \iff \\ \dot{u} + A u = b $$ Then look for $u = e^{-At} v$, which implies $u_0 = v_0$.
This gives $\dot u = -A\,e^{-At} v + e^{-At}\dot v$ and $e^{-At} \dot v = b$, so $\dot v = e^{At} b$ and $$ v = \int\limits_0^t e^{A\tau} b\, d\tau + v_0 $$ and \begin{align} u &= e^{-At}\left( \int\limits_0^t e^{A\tau} d\tau\, b + v_0 \right) \\ &= e^{-At}\int\limits_0^t e^{A\tau} d\tau\, b + e^{-At} u_0 \end{align}