Problem 5 on heat flow handout

Solve the non-linear heat flow problem
$$
\begin{array}{cc}
(uu')' = 0 & 0 < x < \pi \\
u(0) = 0 & u'(\pi) = 1.
\end{array}
$$

This looked like a decent practice problem, so I gave it a shot. Here's what I did:

The equation $(uu')'=0$ can be integrated to give $uu' = c_1$, which is separable. Integrating here gives

$$\frac{u^2}{2} = c_1 x+c_2,$$

which can be solved for $u$, resulting in the general solutions $u(x)=\pm \sqrt{2}\sqrt{c_1x+c_2}$

Putting the factor of $\sqrt{2}$ into the constants, $u(x) = \pm \sqrt{C_1 x + C_2}$. The first condition is then

$$u(0)=\pm \sqrt{0+C_2}=0,$$

so $C_2 = 0$. Then,

$$u(x)=\pm \sqrt{C_1x} = \pm \sqrt{C_1} \sqrt{x} = C\sqrt{x}.$$

Where the constant is re-written for a prettier form. The second condition gives

$$u'(\pi) = \frac{C}{2\sqrt{\pi}}=1,$$

and so $C = 2\sqrt{\pi}$.

The solution is then $u(x) = 2\sqrt{\pi x}$.

Does this check out with you guys? Over the given interval, the function increases at a slowing rate, which seems reasonable to me.

Looks good!