Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

share|cite|improve this answer
Thanks a lot for this!! – Q.matin Nov 27 '12 at 1:09

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.