# Quasilinear 2nd order PDE, apply initial data to general solution

The question involves finding the solution to a partial differential equation.

The general solution that I found was $$u(x,t)=F(x^{2}-t^2 e^{u})$$ and the initial condition is $$u(x,0)=2\ln (x)$$. The issue I've been having is trying to apply that data to the general form.

What I tried was plugging in the data to the general solution to get: $$2\ln (x) = F(x^{2})$$ From there I let $$x^{2} = z$$ and hence $$x=\sqrt{z}$$, which gives me: $$2\ln(\sqrt z ) = F(z)$$

and then I tried taking this and subbing it through to the original solution I had, which gives:

$$u(x,t) = 2\ln (\sqrt{x^{2}-t^2e^{u}})$$

The solution to the problem is $$u(x,t) = \ln (\frac{x^{2}}{1+t^{2}})$$, was wondering if anyone could help point out what i did wrong.

Original PDE is $$\partial_t u + \big(\frac{t}{x}e^u\big)\partial_x u = 0$$ (see problem statement).

If you use the logarithm laws, $$a\ln b=\ln( b^a)$$, you find that $$F(x^2)=\ln(x^2)$$, thus $$F(z)=\ln z$$ and with that $$u=\ln(x^2-t^2e^u)\implies e^u=x^2-t^2e^u\implies e^u=\frac{x^2}{1+t^2}.$$