linear differential equation with ivp

Question

Solve the differential equation:

$$x'(t) = \begin{bmatrix}a&1&0\\0&a&1\\0&0&0\end{bmatrix}x(t)$$ with $x(t)=(x_1(t),x_2(t),x_3(t))$ and $a\in \mathbb R$.

Now I calculated the general solution $\phi(t)=c_1 e^{at}\begin{bmatrix}0\\0\\1\end{bmatrix}+c_2 e^{at}\begin{bmatrix}0\\1\\0\end{bmatrix}+c_3 e^{at}\begin{bmatrix}1\\0\\0\end{bmatrix}$.

And there is a follow-up task: Calculate the solution with $x(0)=\begin{bmatrix}0\\0\\1\end{bmatrix}$

Can anyone help me with that?

Answer 1

When $t=0$, you have

$$\begin{bmatrix}0\\0\\1\end{bmatrix}=\begin{bmatrix}0\\0\\c_1\end{bmatrix}+\begin{bmatrix}0\\c_2\\0\end{bmatrix}+\begin{bmatrix}c_3\\0\\0\end{bmatrix}=\begin{bmatrix}c_3\\c_2\\c_1\end{bmatrix}$$

which means $c_1=1$ and $c_2=c_3=0$.