1. **Problem:** Verify if $u = e^x \sin 2y$ satisfies the partial differential equation $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 11.$$\n\n2. **Step 1: Compute $\frac{\partial^2 u}{\partial x^2}$**\nGiven $u = e^x \sin 2y$, first derivative w.r.t. $x$ is $$\frac{\partial u}{\partial x} = e^x \sin 2y.$$\nSecond derivative w.r.t. $x$ is $$\frac{\partial^2 u}{\partial x^2} = e^x \sin 2y.$$\n\n3. **Step 2: Compute $\frac{\partial^2 u}{\partial y^2}$**\nFirst derivative w.r.t. $y$ is $$\frac{\partial u}{\partial y} = e^x \cdot 2 \cos 2y = 2 e^x \cos 2y.$$\nSecond derivative w.r.t. $y$ is $$\frac{\partial^2 u}{\partial y^2} = 2 e^x \cdot (-2) \sin 2y = -4 e^x \sin 2y.$$\n\n4. **Step 3: Sum the second derivatives**\n$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = e^x \sin 2y - 4 e^x \sin 2y = -3 e^x \sin 2y.$$\n\n5. **Step 4: Check if the sum equals 11**\nThe expression $$-3 e^x \sin 2y$$ is not a constant 11 for all $x,y$.\n\n**Conclusion:** The function $u = e^x \sin 2y$ does not satisfy the PDE $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 11.$$\n\n**Note:** The PDE is a Poisson equation with constant right side 11, but $u$ given is not a solution to it.