Find fundamental matrix of a 2x2 matrix with rank 1

Question

$$ x'(t) = \left[\begin{array}{cccc}0&1\\0&t\end{array}\right]x(t)$$

I am having trouble computing the fundamental matrix. I get: $$ x_1(t) = x_2(0)*exp(0.5t^2) $$ $$ x_2(t) = x_2(0)*exp(0.5t^2) $$ where $$ x_2(0) $$ is the initial condition.

I choose the initial conditions to be $$ x(0) = \left[\begin{array}{cccc}1\\ 2\end{array}\right], x(0) = \left[\begin{array}{cccc}0\\ 1\end{array}\right]$$

so I get fundamental matrix to be $$ \left[\begin{array}{cccc}2exp(0.5t^2)& exp(0.5t^2)\\ 2exp(0.5t^2)& exp(0.5t^2)\end{array}\right]$$

but this matrix cannot be inverted (or I don't know how) to compute the state transition matrix.

So what did I do wrong?

Answer 1

$x_2(t)=x_2(0)e^{\frac12t^2}$ and $x_1'(t)=x_2(t)$ implies $$x_1(t)=x_1(o)+x_2(0)\int_0^t e^{\frac12x^2}dx$$