# Change of variables on autonomous systems solutions

#### GlassBones

1. The problem statement, all variables and given/known data

Given that $x=\phi (t)$, $y=\psi(t)$ is a solution to the autonomous system $\frac{dx}{dt}=F(x,y)$, $\frac{dy}{dt}=G(x,y)$ for $\alpha < t < \beta$, show that
$x=\Phi(t)=\phi(t-s)$, $y=\Psi(t)=\psi(t-s)$
is a solution for $\alpha+s<t<\beta+s$ for any real number s.

2. Relevant equations

3. The attempt at a solution
I noticed $\alpha+s<t<\beta+s \equiv \alpha<t-s<\beta$. I'm thinking to do change of variables. But don't really know how to do that.

Notation wise does this makes sense $\frac{dx}{d(t-s)}$.

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#### Orodruin

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Hint: Chain rule for derivatives.

"Change of variables on autonomous systems solutions"

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