Answer
$xX''(x)-\lambda X(x) = 0$ and $T'(t)+\lambda T(t) = 0$
Work Step by Step
Given the partial differential equation
$xu_{xx} + u_{t} = 0$,
we first try the method of separation of variables toward a solution:
Let $u(x,t) = X(x)T(t)$.
Then,
$x(X(x)T(t))_{xx} + (X(x)T(t))_{t} = 0$;
rewriting,
$xT(t)X''(x) = -X(x)T'(t)$,
$\frac{xX''(x)}{X(x)} = -\frac{T'(t)}{T(t)} = \lambda$.
This give two ordinary differential equations,
$xX''(x) - \lambda X(x) = 0$ and $T'(t) + \lambda T(t) = 0$.
