Question $1$

Consider the initial boundary value problem with Neumann boundary conditions.

$x{u}_x+y{u}_y=x^{2}y,x,y>0,u(1,y)=y{u}_t=u_{}00$

$u_x(0,t)=0,u_x(,t)=0,t>0$

$u(x,0)=x,0$

Use the separation $of$ variable method $to$ solve this problem.

Write a program $in$ Mathematica $or$ MATLAB $in$ order $to$ plot the sum $of$ the $70$ first terms $of$ the solution $asan$ approximation $to$ this solution.

Solve the partial differential equation: $x{u}_x+y{u}_y=x^{2}y,x,y>0,u(1,y)=y$

Write a program $to$ sketch this solution using Mathematica $or$ MATLAB.