Let A be a constant matrix. Suppose 𝐮(t) solves the initial value problem 𝐮=A 𝐮, 𝐮(0)=𝐛. Prove t

step 1 Okay, so we have the system x -primes equal to a of t, x -vector, x -prime vector as well. And w

Let A be a constant matrix. Suppose 𝐮(t) solves the initial...

Let $A$ be a constant matrix. Suppose $\mathbf{u}(t)$ solves the initial value problem $\mathbf{u}=A \mathbf{u}$, $\mathbf{u}(0)=\mathbf{b}$. Prove that the solution to the initial value problem $\dot{\mathbf{u}}=A \mathbf{u}, \mathbf{u}\left(t_0\right)=\mathbf{b}$, is equal to $\overline{\mathbf{u}}(t)=\mathbf{u}\left(t-t_0\right)$. How are the solution trajectories related?

Key Concepts

Autonomous Systems

An autonomous system is a differential equation in which the independent variable (often time) does not appear explicitly in the system itself. This property implies that if a function is a solution to the system, then a shifted version of that function (translated in time) is also a solution. In the context of linear systems with constant matrices, this means that the behavior of the solution is invariant under time translation, leading to a family of solutions corresponding to shifted initial times.

Time Translation Invariance

Time translation invariance refers to the characteristic of a differential equation where shifting the time variable does not change the form of the equation. For linear systems with constant coefficients, this property ensures that if u(t) is a solution with a given initial condition, then u(t - t0) is also a valid solution that satisfies the modified initial condition at t0. This invariance is at the heart of proving that the solution trajectories are simply shifted versions of one another.

Initial Value Problems and Uniqueness

Initial value problems (IVPs) involve finding a solution to a differential equation that satisfies given conditions at a specified initial time. The uniqueness theorem for linear systems guarantees that, under appropriate conditions, the IVP has a unique solution. This uniqueness underpins the argument that if a time-translated solution satisfies the same differential equation and meets the new initial condition, then it must coincide with the unique solution corresponding to that condition.

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