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    Quasi-Linear Partial Differential Equation

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    Show that the solution u of the quasi-linear partial differential equation

    u_y + a(u)u_x = 0.

    With initial condition u(x,0) = h(x) is given implicitly by

    u(x,y) = h(x-a(u)y).

    Show that the solution becomes singular for some positive y, unless a(h(s)) is a nondecreasing function of s.

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    Solution Summary

    A quasi linear partial differential equation (PDE) is investigated in this detailed solution which illustrates how substitution can be used to solve for this derivation. The response is enclosed within an attached pdf. file.