In general for a system of IVP \begin{align*} \dot{x}=Ax+b(t) && x(t_0)=x_0 \end{align*} we can write the solution as \begin{align*} x(t)=\Phi(t) \Phi(t_0)x_0+\Phi(t)\int _{t_0}^t\Phi^{-1}(s)b(s) \end{align*} What I am trying to understand is how is the formula the same as would be obtained using the method of Variation of Parameters for a 2nd order linear non-homogeneous ODEs.

What I was thinking is to consider a general second order ODE $$y''+P(t)y'+Q(t)y=b(t)$$ Then my thought was to write it as a system of equations to be able to batch this to the formula.

\begin{align*} \begin{bmatrix} x'_1\\ x'_2 \end{bmatrix} =& \begin{bmatrix} 0 &1\\ -P(t) &-q(t)\\ \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \end{bmatrix} +& \begin{bmatrix} 0\\ b(t) \end{bmatrix} \end{align*}

Then I am really not sure. I mean I do not need a solid proof more of what I want is to see how are the solution using variation of parameters the same as the formula we have using the fundamental matrix.


Your Answer

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

Browse other questions tagged or ask your own question.