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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?

GoodDeeds
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1 Answers1

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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 $u=X/Y$.

mike
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  • Thanks a lot! It is very kind of you and I think you are right about the substitution. But I think something in my calculation is wrong, because I substitute X and y by function of x and t in the right solution (4x-y)^2/16, it is not equal to z. Can you help me check what is wrong with my calculation? –  Sep 13 '14 at 04:13
  • I checked your math but I was not able to find any problem. Sorry, that is all I can help. – mike Sep 13 '14 at 04:59
  • Yeah, I checked it many times today and I also cannot find a problem. But I still cannot get to the right answer from the calculation. Anyway, thanks very much! –  Sep 13 '14 at 18:53