### Author Topic: Overdetermined systems  (Read 767 times)

#### Ioana Nedelcu

• Full Member
•   • Posts: 29
• Karma: 3 ##### Overdetermined systems
« on: January 10, 2018, 09:34:55 PM »
I'm having some trouble solving the overdetermined systems in the homework assignment.

\begin{align} & u_{xx}=2xy \label{A}\\ &u_{y}=x^2 \label{B} \end{align}
Solving the last equation gives $u = x^2y + f(x)$

Then using the first equation: $$u_{xx} = 2xy = 2y + f_{xx}(x)$$ Then solving for $f(x)$ gives $f(x) = x^3y/3 -x^2y$ so the overall solution is $$u = x^3y/3$$

However, this clearly doesn't satisfy the partial derivative equation with respect to $y$.

Just wondering what I'm doing wrong? I haven't had much experience with systems, even in ODE

#### Victor Ivrii ##### Re: Overdetermined systems
« Reply #1 on: January 11, 2018, 04:59:19 AM »
1) THis is problem not from the homework: it was added just few days ago and homework description from "all problems" was changed to "problems 1--5" at http://www.math.toronto.edu/courses/apm346h1/20181/lectures.html (Week 2). We do not study such systems in the course.

2) It contained an error (fixed now) which did not make it senseless.

3) For overdetermined systems the usual case is as in
http://forum.math.toronto.edu/index.php?topic=882.30
a) Solution does not exist unless right hand expression satisfies certain compatibility conditions (depending on the system).
b) If solutions exist then there are less of them than in the case of determined system.

The simplest example (which you actually studied in Calculus II)
\left\{ \begin{aligned} &u_x=f,\\ &u_y=g. \end{aligned}\right.
Compatibility condition $f_y=g_x$ and under this condition $u$ is defined up to a constant (so the general solution does not contain an arbitrary function of one variable).

4) Another example is the Maxwell system. There are 6 unknown (components of electric $\mathbf{E}$ and magnetic $\mathbf{H}$ fields) and 8 equations. The right-hand expressions may contain densities of the charges $\rho$ and currents $\mathbf{j}$.

« Last Edit: January 11, 2018, 05:03:46 AM by Victor Ivrii »