1. (a) Form the partial differential equation of the following
(i) $z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$
(ii) $z = f(x + ay) + g(x - ay)
(b) Use Lagrange's Method to find the general solution of
$\frac{y^2z}{x} \frac{\partial z}{\partial x} + xz \frac{\partial z}{\partial y} = y^2$.
(c) Solve the heat conduction equation
$\frac{\partial^2 u}{\partial x^2} = \frac{1}{2} \frac{\partial u}{\partial t}$
over $0 < x < 3$, $t>0$ for the boundary conditions
$u(0,t) = u(3,t) = 0$
and the initial condition
$u(x, 0) = 5 \sin 4\pi x$.
