Change of variables on autonomous systems solutions

  • #1
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Homework Statement



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.

Homework Equations




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)}##.
 

Answers and Replies

  • #2
Orodruin
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Hint: Chain rule for derivatives.
 

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