-1
$\begingroup$

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.

$\endgroup$
-1
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ It doesn't work like that for non-autonomuous systems. $\endgroup$ – Gustave Jun 12 at 7:31
  • $\begingroup$ What doesn't work like what? $\endgroup$ – mjw Jun 12 at 11:50
  • $\begingroup$ The expression seems to satisfy the equation, no? $\endgroup$ – mjw Jun 16 at 4:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.