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Let A be a constant matrix. Suppose u(t) solves the inital value problem $\dot u = Au$, $u(0) = b$. Prove that the solution to the inital value problem $\dot u = Au$, $u(t_0) = b$ is equal to $\hat u = u(t -t_0)$. How are the solution trajectories related?

I am not good with proofs. Never have I done it before prior of taking this differential equation class. Can anyone show me how to prove this?

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up vote 3 down vote accepted

The solution to $\dot{u}(t) = Au(t)$ with $u(0) = b$ is $u(t) = \exp(tA)b$. (This can be verified by differentiation.) The solution to $\dot{\hat{u}}(t)=A\hat{u}(t)$ with $\hat{u}(t_{0})=b$ is $\hat{u}(t) = \exp((t-t_{0})A)b$. But $\hat{u}(t) = \exp((t-t_{0})A)b =\exp(t'A)b =u(t') = u(t-t_{0})$.

The solution trajectories are related by a time-translation.

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Thanks a lot for this!! – Q.matin Nov 27 '12 at 1:09

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