I'm stuck on problem 3.4.1 of Berkeley Problems in Mathematics. It states the following:
Consider the system of differential equations:
$\frac{dx}{dt} = y + tz$
$\frac{dy}{dt} = z + t^2x$
$\frac{dz}{dt} = x + e^ty$
Prove that there exists a solution defined for all $t \in [0, 1]$, such that $$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \quad \begin{pmatrix} x(0) \\ y(0) \\ z(0) \end{pmatrix} \quad = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} $$
and also $\int_{0}^{1} (x(t)^2 + y(t)^2 + z(t)^2) dy = 1$.
Now I understand that the matrix implies that $x, y, z$ are $0$ at $t = 0$, and that if we could prove that $x, y, z$ are continuously differentiable then we would be done by the existence/uniqueness theorem.
I don't know how to proceed from here though, is the integral simply a hint for finding functions $x, y, z$ or is there some method I am missing here.
The linear system is not regular, its solution set is determined as $z(0)=x(0)$, $y(0)=-2x(0)$.
As the ODE system is linear, you only need the solution for $x(0)=1$, all others are multiples of it, and you can determine a scaling factor so that also the integral condition is satisfied.