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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 tex2html_wrap_inline139 and tex2html_wrap_inline141 which satisfy the two equations. You can think about the system geometrically:

Suppose that the tex2html_wrap_inline143 , tex2html_wrap_inline145 plane is the surface of the sea, and the velocity vector of the water at any point tex2html_wrap_inline147 is tex2html_wrap_inline149 A solution tex2html_wrap_inline151 is a parametrized curve in tex2html_wrap_inline153 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 tex2html_wrap_inline147 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 tex2html_wrap_inline161 )


A pair of functions tex2html_wrap_inline139 and tex2html_wrap_inline141 is a solution of (1) if and only if the vector valued function tex2html_wrap_inline167 is a solution of (2).

  1. Show that the pair tex2html_wrap_inline169 , tex2html_wrap_inline171 is a solution of (1).
  2. Let A be a tex2html_wrap_inline175 matrix. Show that if the vector tex2html_wrap_inline177 is an eigevector of A with eigenvalue tex2html_wrap_inline181 , then the vector valued function tex2html_wrap_inline183 is a solution of the equation tex2html_wrap_inline185

    Hint: Simply plug tex2html_wrap_inline183 for tex2html_wrap_inline147 on both sides of the equation. Recall the Chain Rule: tex2html_wrap_inline191

  3. Find two linearly independent vectors tex2html_wrap_inline193 and tex2html_wrap_inline195 and numbers tex2html_wrap_inline197 and tex2html_wrap_inline199 , such that tex2html_wrap_inline201 and tex2html_wrap_inline203 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 tex2html_wrap_inline205 and tex2html_wrap_inline207 .

  4. Find the path of the raft which passes through the point tex2html_wrap_inline209 in the plane at time t=0. Hint: Plug 0 for t in (3), set it equal to tex2html_wrap_inline209 and solve for tex2html_wrap_inline205 and tex2html_wrap_inline207 .

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Eyal Markman
Thu Dec 7 17:14:57 EST 2000