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Math Help - rewrite implicit solution (get u out of an exponent)

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

    Hey guys. I'm working on the IVP

    u_y=xuu_x, u(x,0)=x,

    and given the implicit solution 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 u=u(x,y). (This is the reason this is in the analysis section.)

    Any help would be much appreciated. Thanks!
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  2. #2
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    Re: rewrite implicit solution (get u out of an exponent)

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

    u_y=xuu_x, u(x,0)=x,

    and given the implicit solution 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 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 f(x)= xe^x.

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