Discussion
For the differential equation given below, which one of the following options is correct?
\(\frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}} = 0\) \(\quad 0 \le x \le 1,\ 0 \le y \le 1\)
Answer: B
The solution is \(u = e^{x}\sin y\).
Check Laplace's equation term by term: \(u_{xx} = e^{x}\sin y\) and \(u_{yy} = -e^{x}\sin y\), so \(u_{xx} + u_{yy} = 0\) everywhere.
For \(e^{x+y}\) the sum is \(2e^{x+y}\); for \(\sin x\sin y\) it is \(-2\sin x\sin y\); for \(\cos x\cos y\) it is \(-2\cos x\cos y\). None of these vanishes for all \(x\) and \(y\), so none is a solution.
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