I'm trying to solve this question from Salsa's PDE in action:
If we let $u = t^{\alpha}U(\xi)$ , where $\xi = x/t^{\beta}$, and
$$u_t = t^{\alpha-1}\left(\alpha U(\xi) - \beta\ \xi U'(\xi)\right)$$
$$ u_{xx} = t^{\alpha - 2 \beta} U''(\xi)$$
then for part a), letting $\beta = 1/2$, we get:
$$u_t - u_{xx} = t^{\alpha-1}\left(\alpha U(\xi) - \frac{\xi}{2} U'(\xi) - U''(\xi)\right) = 0$$
and for part b, letting $\alpha = 1$ and $\beta = 1/2$
$$u_t - u_{xx} = 1\cdot \left( U(\xi) - \frac{\xi}{2} U'(\xi) - U''(\xi)\right) = 1$$
and same thing for part $c$.
My question is how to prove that the given parameters $\alpha$ and $\beta$ are appropriate for each $f(x)$. I don't see how