(Solved): Separation of variables Consider the linear partial differential equation x22u=yyu ...

Separation of variables Consider the linear partial differential equation $\frac{{?}^{2}u}{?x^2}=y\frac{?u}{?y}?y^2\frac{{?}^{2}u}{?y^2}\text{ for }0<x<?,0<y<1$ with the conditions $u(0,y)=u(?,y)=u(x,0)=0\text{ and }{u}_x(x,1)=12\cos (4x)+14\cos (7x)$ 1. Derive two ODEs (one depending on $x$ and the other on $y$ ) by seeking a separation solution of the form $u(x,y)=X(x)Y(y)$, using $k$ to denote the separation constant. 2. Find the non-trivial solutions of the ODE that depends on $x$, satisfying the conditions $u(0,y)=u(?,y)=0$, and determine an expression for the values of the separation constant $k$. 3. Determine the solutions for the ODE in $y$ which satisfy the condition $u(x,0)=0$. Hint: Use the Euler-Cauchy solution method 4. Use the superposition principle for linear PDEs to write down the solution for $u(x,y)$ that satisfies the boundary conditions $u(0,y)=u(?,y)=u(x,0)=0$. 5. Determine the solution of $u(x,y)$ that satisfies the boundary condition $u_x(x,1)=12\cos (4x)+14\cos (7x)$