The system of two equations

 

is an example of a linear system of ordinary differential equations (because it involved derivatives). A solution of (1) is a pair of two differentiable functions and which satisfy the two equations. You can think about the system geometrically:

Suppose that the , plane is the surface of the sea, and the velocity vector of the water at any point is A solution is a parametrized curve in indicating the path of a raft floating on the surface of the sea. The system (1) reads:

If the solution passes through the point with coordinates at time t, then its velocity vector at that time is equal to the velocity vector of the water at that point.

The system of equations can be written in matrix form: (where x' stands for )

 

A pair of functions and is a solution of (1) if and only if the vector valued function is a solution of (2).

1. Show that the pair , is a solution of (1).
2. Let A be a matrix. Show that if the vector is an eigevector of A with eigenvalue , then the vector valued function is a solution of the equation

   Hint: Simply plug for on both sides of the equation. Recall the Chain Rule:

3. Find two linearly independent vectors and and numbers and , such that and are solutions of (2).

   It is easy to check that the sum of two (vector valued) solutions of (2) is again a solution of (2). Conclude that

    

   is a solution for any choice of constants and .

4. Find the path of the raft which passes through the point in the plane at time t=0. Hint: Plug 0 for t in (3), set it equal to and solve for and .

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