Explicit formula of a fundamental matrix

Question

I'm trying to solve the following differential system explicitly $$\left( \begin{array}{c} y \\ z% \end{array}% \right) ^{\prime }=\left( \begin{array}{cc} 0 & b(t) \\ \delta b(t) & a(t)% \end{array}% \right) \left( \begin{array}{c} y \\ z% \end{array}% \right) ,\text{ }\delta \in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} $$ Can we find an explicit formula of the solutions? Thank you.

Answer 1

The matrix equation $$\frac{d\mathbf{x}}{dt} = A(t) \mathbf{x}$$ has solution

$$\displaystyle \mathbf{x}(t) = \mathbf{x}(0) e^{\int_0^t A(\tau) d\tau}. $$

$$\displaystyle \mathbf{x} = \begin{pmatrix} y\\z \end{pmatrix}, \quad e^{\int_0^\tau A d\tau} = I + \int_0^t d\tau \begin{pmatrix} 0 & b(t) \\ \delta b(t) & a(t) \end{pmatrix} + \frac{1}{2!} \left\{ \int_0^t d\tau \begin{pmatrix} 0 & b(t) \\ \delta b(t) & a(t) \end{pmatrix} \right\}^2 + \cdots$$

You can compute the elements of $\mathbf{x}(t)$ by summing the series.