Higher Order Partial Derivatives — Question 10

Question 10 – Solution

Strategy. Equality for every $x$ forces independent trigonometric coefficients to vanish.

Step 1: Differentiate $$u_{xx}=e^{ax+by}[(a^2-c^2)\cos(cx)-2ac\sin(cx)],$$ $$u_{yy}=b^2e^{ax+by}\cos(cx).$$ Thus the sum is $$e^{ax+by}[(a^2+b^2-c^2)\cos(cx)-2ac\sin(cx)].$$

Step 2: Nonzero $c$ If $c\ne 0$, independence of sine and cosine gives $$a^2+b^2-c^2=0,\qquad ac=0.$$ Thus $a=0$ and $b=\pm c$.

Step 3: Edge case If $c=0$, the condition is $a^2+b^2=0$, hence $a=b=0$. This is already included below: $$\boxed{(a,b,c)=(0,c,c)\ \text{or}\ (0,-c,c),\quad c\in\mathbb R}.$$ Substitution makes both coefficients vanish, verifying sufficiency.