# Thread: rewrite implicit solution (get u out of an exponent)

1. ## rewrite implicit solution (get u out of an exponent)

Hey guys. I'm working on the IVP

$\displaystyle u_y=xuu_x$, $\displaystyle u(x,0)=x$,

and given the implicit solution $\displaystyle x=ue^{-yu}$.

(This is from Fritz John's Partial Differential Equations, 4th Ed, exercise 1.6.1b p18.)

It's easy to show that this is indeed a solution, but now I am expected (I think) to put it into explicit terms, i.e. in the form $\displaystyle u=u(x,y)$. (This is the reason this is in the analysis section.)

Any help would be much appreciated. Thanks!

2. ## Re: rewrite implicit solution (get u out of an exponent)

Originally Posted by hatsoff
Hey guys. I'm working on the IVP

$\displaystyle u_y=xuu_x$, $\displaystyle u(x,0)=x$,

and given the implicit solution $\displaystyle x=ue^{-yu}$.

(This is from Fritz John's Partial Differential Equations, 4th Ed, exercise 1.6.1b p18.)

It's easy to show that this is indeed a solution, but now I am expected (I think) to put it into explicit terms, i.e. in the form $\displaystyle u=u(x,y)$. (This is the reason this is in the analysis section.)

Any help would be much appreciated. Thanks!
If you really must write u as an explicit function of x and y, you will have to use the "Lambert W function" which is defined as the inverse function to $\displaystyle f(x)= xe^x$.

Let v= yu so that u= v/y. Then the equation becomes $\displaystyle x= \frac{v}{y}e^{-v}$ so that $\displaystyle xy= ve^v$ and taking the W function of both sides, $\displaystyle W(xy)= v= yu$ and, finally, $\displaystyle u= \frac{W(xy)}{y}$.