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During my course in mathematical methods for physics, my professor introduced us to the "method of fixed point" for studying non-homogeneous systems of odes, odes and pdes. Although I think I understood how to use it, he wasn't so clear about WHERE to use it. Let me make some examples: $$\begin{align}&\left\{\begin{matrix}\dot{x}=3x-2y+f(t)\\\dot{y}=-2x+3y\end{matrix}\right.& f(t) = \left\{\begin{matrix}1\;\;\;\;0<t<1\\0\;\;\;\;t>1\end{matrix}\right.\end{align}$$$$\left\{\begin{matrix}\partial_{t}u = D\partial^{2}_{xx}u + f(x) \\ f(x) = \frac{d^2}{dx^2}e^{-ax^2}\\u(x,0)=e^{-ax^2}\end{matrix}\right.$$ $$\left\{\begin{matrix}\partial_{t}\phi = D\partial^{2}_{xx}\phi + \sin(2x)\\ 0<x<\pi\\ \phi(x,0) = \sin(x)+3\sin(8x) \\ \phi(0,t)=\phi(\pi,t)=0\end{matrix}\right.$$

Two of these problems, mainly the last two, I know for a fact that can be solved around their fixed points: $$\tilde{u}(x,t) = u(x,t) + {1\over D}e^{-ax^2}$$ and $$\tilde{\phi}(x,t) = \phi(x,t) + {1\over{4D}}sin(2x)$$ (even if the second one is pretty unnecessary). But for the first one I'm pretty confused. I know that that system of odes can be rewritten in the form $$\dot{\underline{x}} = \hat{A}\underline{x}+\underline{f}$$ which has solution $$\underline{x}(t) = \underline{x}(0)e^{\hat{A}t}+\int_{0}^{t}e^{\hat{A}(t-t')}\underline{f}(t')dt'$$ and from what I understood during classes I cannot study a non-homogeneous in the fixed point if the external force is a function of time, but in my notes I found that our professor studied the fist system with a variable $$\underline{z} = \underline{x}-\hat{A}^{-1}\underline{f}$$ which seems to me like a fixed point! Probably I'm confusing a lot but I cannot wrap my head around it. So coming to my question: is what I said in bold true or am I going crazy? Is it possible to study a problem like this one problem like this one with the fixed point method? Thank you very much in advance to anyone that can give me some kind of explanation. Sadly this course is very interesting but the limited time available makes it very hard to explain everything in detail.

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