Past exam of the differential equations and applications course of the University of Oxford / 2010 / B / 6 / a

For $\eta =x/y^{\beta }$ the chain rule gives $u_y=-\beta \eta f^{\prime }/y$, $u_x=y^{-\beta }{f}^{\prime }$ and $(u{u}_x{)}_x=y^{-2\beta }[f{f}^{\prime \prime }+(f^{\prime }{)}^2]$. Multiplication of the equation by $y$ givesA nontrivial scale-invariant balance therefore selects $\boxed{\beta =1/2}$ and yieldsIts left side is the derivative of $f{f}^{\prime }+\eta f/2$. The boundary conditions give $f(0)=1$ and $f^{\prime }(0)=0$, so the integration constant is zero:On the positive branch before the first zero, divide by $f$ and integrate, obtaining $f=1-{\eta }^2/4$. Its first positive zero is ${\eta }_F=2$. HenceA direct check gives $u_y=x^2/(4y^2)$ and $(u{u}_x{)}_x+u/(2y)=x^2/(4y^2)$. The zero extension past this front is also a natural weak compact-support continuation: $u$ is continuous and the degenerate diffusive flux $u{u}_x$ tends to zero at the front. This is a source-driven porous-medium similarity front.