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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.

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