What is wrong in solving this pde?

Question

I solved the first order pde and I found it is impossible to express $x$ and $t$ using $X$ and $y$, so I cannot get the solution $u$ from $z$. But the right answer is pretty simple. It is $\frac{(4x-y)^2}{16}$. Can anyone help me find what is wrong with my calculation?

Answer 1

We start from the last two expressions you obtained:

$$X(x,t)=x\left(1+\frac{t}{2}\right)\tag{1}$$ $$Y(x,t)=2x(e^t-1)\tag{2}$$

(1)/(2) leads to:

$$u=X/Y=\frac{2+t}{4(e^t-1)}\tag{3}$$

$$t = -2 - 4u - W(-4ue^{-2 - 4u}) \tag{4}$$

Where W(z) is called Lambert W(z) function, which is solution of function $z=We^W$.

You can then substitute (4) back into (1) to solve x in terms of $X$ and $u=X/Y$.