In your answers below, for the variable $\lambda$ type the word lambda; for the derivative $\frac{d}{dx} X(x)$ type X' ; for the double derivative $\frac{d^2}{dx^2} X(x)$ type X''; etc.

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Separate variables in the following partial differential equation for $u(x,t)$: $$tu_x-t^{2}u_{xt}+x^{3}u_{tt}-x^{3}u = 0$$
    $=$ $= \lambda$

• DE for $X(x)$: $= 0$
  
  
• DE for $T(t)\,$: $= 0$

    (Simplify your answers so that the highest derivative in each equation is positive.)