Lemma 12.3.6.label The Cauchy problem in Lemma 12.3.4 is equivalent to a Cauchy problem of the same form with $A_{j}$ and $B$ independent of $t$ and $\Phi=0$.
Proof. Set $U(x,t)\define Y(x,t)-\Phi(x)$ the system becomes
\begin{align*}\begin{cases}\pd_{t}U=\sums{n-1}{\el=1}A_{\el}(x,t,U+\Phi)\pd_{\el} U+B(x,t,U+\Phi)\\ U(x,0)=0\end{cases}\end{align*}
We eliminate $t$ by adding component $u^{0}$ to $U,\begin{pmatrix}u^{0}\\U\end{pmatrix}$ then
\begin{align*}\pd_{t}u^{0}=1&&u^{0}(x,0)=0\end{align*}
[1]$\square$
- This is a simple trick in dynamics to make nonautonamous to autonamous equation by making time into a variablekeyboard_return
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