# How did they get an analytical solution to this second-order nonlinear PDE?

A (non-math) paper I'm working through presents the following differential equation (and solution) in a casual way:

$$0 = \frac{(1-\gamma)^{2}}{\gamma} e^{-\rho t} \bigg[ \frac{e^{\rho t} J_{x}}{\beta} \bigg]^{\frac{\gamma}{\gamma-1}} + J_{t} + [(1-\gamma) \eta / \beta + r x] J_{x} - \frac{J_{x}^{2}}{J_{xx}} \frac{(\alpha - r)^{2}}{\alpha \sigma^{2}},$$ subject to $$J(x,T)=0$$.

A solution is $$J(x,t) = \delta \beta^{-\gamma} e^{-\rho t} \bigg[ \frac{\delta(1-e^{-\left( \frac{\rho - \gamma \nu}{\delta}\right)(T-t)})}{\rho - \gamma \nu} \bigg]^{\delta} \bigg[ \frac{x}{\delta} + \frac{\eta}{\beta r}(1-e^{-r(T-t)}) \bigg]^{\gamma},$$ where $$\delta \equiv 1 - \gamma$$ and $$\nu \equiv r + (\alpha - r)^{2}/2 \delta \sigma^{2}$$.

It seemed rather matter-of-fact, but I would have had no idea how to solve (what appears to be) such a complicated PDE. I've been looking through Polyanin's Handbook of Nonlinear Partial Differential Equations, but I haven't (yet) been able to find anything.

Any ideas on how they did it? (Or recommendations for relevant references?)

The second derivative $$J_{xx}$$ in the denominator is unusual, but maybe some progress is possible. The explicit time exponentials can be removed by substituting $$J(x,t) = e^{(\gamma-1)\rho t}Y(x,t).$$ I won't keep track of the many constants, but you get an equation of the form $$c_1 Y_{x}^{\gamma/(\gamma-1)} +c_2Y+c_3Y_t +(c_4+rx)Y_x-c_5\frac{Y_x^2}{Y_{xx}} = 0.$$ It is still a strange equation. But maybe it suggests powers of something plus $$x$$. So try $$Y(x,t) = (h(t)+x)^\gamma.$$ We get $$c_1\gamma^{\gamma/(\gamma-1)}(h+x)^\gamma + c_2(h+x)^\gamma+c_3\gamma(h+x)^{\gamma-1}h_t$$ $$+ (c_4+rx)\gamma(h+x)^{\gamma-1} -c_5\frac{(\gamma(h+x)^{\gamma-1})^2}{\gamma(\gamma-1)(h+x)^{\gamma-2}} = 0.$$ Factor out $$(h+x)^\gamma$$ to get $$c_1\gamma^{\gamma/(\gamma-1)} + c_2+\gamma\frac{c_3 h_t+c_4+rx}{h+x} -c_5\frac{\gamma}{\gamma-1} = 0.$$ It is easy to find a function $$h(t)$$ satisfying $$c_3h_t+c_4 = rh.$$ Then you are left with the question whether the constants $$c_1\gamma^{\gamma/(\gamma-1)} + c_2+\gamma r -c_5\frac{\gamma}{\gamma-1} = 0?$$ If so then it seems that you have a solution. I have not tried to see whether it is the same as the one proposed in the paper, or even check the intial value.