College Math Teaching

March 6, 2010

Why We Shouldn’t Take Uniqueness Theorems for Granted (Differential Equations)

Filed under: differential equations, partial differential equations, uniqueness of solution — collegemathteaching @ 11:07 pm

I made up this sheet for my students who are studying partial differential equations for the first time:

Remember all of those ”existence and uniqueness theorems” from ordinary differential equations; that is theorems like: “Given

y^{\prime }=f(t,y) where f is continuous on some rectangle
R=\{a<t<b,c<y<d\} and (t_{0},y_{0})\in R, then we are guaranteed at least one solution where y(t_{0})=y_{0}. Furthermore, if \frac{\partial f}{\partial y} is continuous in R then the solution is unique”.

Or, you learned that solutions to
y^{\prime \prime }+p(t)y^{\prime}+q(t)y=f(t), y(t_{0})=y_{0}, \ y^{\prime}(t_{0})=y_{1} existed and were unique so long as p,q, and f were continuous at t_{0}.

Well, things are very different in the world of partial differential
equations.

We learned that u(x,y)=x^{2}+xy+y^{2} is a solution to xu_{x}+yu_{y}=2u
(this is an easy exercise)

But, can attempt a solution of the form u(x,y)=f(x)g(y).
This separation of variables technique actually works; it is an exercise to see that u(x,y)=x^{r}y^{2-r} is also a solution for all real r!!!

Note that if we wanted to meet some sort of initial condition, say, u(1,1)=3, then u(x,y)=x^{2}+xy+y^{2}, and u(x,y)=3x^{r}y^{2-r} provide an infinite number of solutions to this problem. Note that this is a simple, linear partial differential equation!

Hence, to make any headway at all, we need to restrict ourselves to studying very specific partial differential equations in situations for which we do have some uniqueness theorems.

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