# Finding example function from nonlinear differential equation

I would like to find a twice continuously differentiable function $$f(x_1,x_2): \mathbb{R}^2 \rightarrow \mathbb{R}$$. Denote $$f_1, f_2, f_{11}, f_{12}, f_{22}$$ its first and second order partial derivatives with respect to $$x_1, x_2,\dots$$. $$f$$ needs to fulfill the following equation $$$$1 - \exp(f) = \frac{f_1}{f_2}a$$$$

where $$a$$ is a given constant. There are no further constrains on the function. I tried to exploit the fact that $$f_{12} = f_{21}$$. Differentiating the equation above on both sides separately with respect to $$x_1$$ and $$x_2$$ and dividing yields:

\begin{align} 2f_{12} = f_{11}\frac{f_2}{f_1} + f_{22}\frac{f_1}{f_2} \end{align}

But I do not know how to proceed from here. Thank you for any help and suggestions.

• In the original equation try the substitution $f = \log (1 + g)$. You end up with something very close to the inviscid Burgers’ equation for $g$. Mar 2, 2021 at 14:17

As @A rural reader suggested, substituting $$f=\ln(1+g)$$ yields the inviscid Burgers' equation: $$-g = \frac{g_1}{g_2} a \Leftrightarrow a g_1 + g~ g_2 = 0.$$ Solutions to this equation have to be classified through an initial condition, e.g. $$g(0,x) = F(x)$$ which has the implicit solution: $$g(x_1,x_2) = F\left(x_2-\frac{x_1}{a}\cdot g \right) \Rightarrow g(x_1,x_2) = \cdots \Rightarrow f(x_1,x_2) = \ln(1+g).$$
A typical example for this would be $$F(x) = x$$, which leads to $$g(x_1,x_2) = \frac{ax_2 }{a+ x_1 } \Rightarrow f(x_1,x_2) = \ln\left( 1 + \frac{ax_2 }{a+ x_1 } \right).$$