Higher Order Partial Derivatives — Question 9

Question 9 – Solution

Strategy. Substitute the separated trial form and compare its common factors.

Step 1: Derivatives $$u_{tt}=-A\omega^2\cos(2x)\cos(\omega t),\qquad u_{xx}=-4A\cos(2x)\cos(\omega t).$$

Step 2: Match For $A\ne 0$, equality for every $(x,t)$ requires $$-\omega^2=9(-4),\qquad\omega^2=36,$$ so $\boxed{\omega=\pm 6}$.

Step 3: Verify Directly, $u_{tt}=-36u=9(-4u)=9u_{xx}$. Because $\cos(-6t)=\cos(6t)$, the signs produce the same function. If $A=0$, the zero function satisfies the equation for every $\omega$.