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    Could someonexplain to me how they got from the "*****" to the next stage of this proof? Thanks

    Q.Given that x=e^u, where u is a function of x, show that:
    x^2.d^2.y/dx^2 = [d^2.y/du^2] - dy/du

    d^2.y/du^2 = [d/du].(dy/du)
    = [d/du].(e^u.dy/dx) *****
    = e^u.dy/dx + e^u.[d^2.y/dx^2]. [dx/du]
    = dy/du + x^2.[d^2.y/dx^2]

    Since dx/du = e^u and e^u=x

    x^2.[d^2.y/dx^2] = [d^2.y/du^2] - dy/du

    (Sorry the layouts not v.easy to understand - I think UKL should have special function buttons for us Mathematicians - or at least for those of us trying to be mathematicians!!!!) Thanks

    d^2.y/du^2 = [d/du].(dy/du)
    = [d/du].(e^u.dy/dx) *****

    ok, from this point, you're simply using the product rule. viz.

    d(uv)/dx = u.dv/dx + v.du/dx

    ( the u above is different from the u that ur using

    [d/du].(e^u.dy/dx) = (e^u).[d/du](dy/dx) + (dy/dx).[d/du](e^u)
    [d/du].(e^u.dy/dx) = (e^u).d²y/dx².dx/du + dy/dx.e^u
    [d/du].(e^u.dy/dx) = e^u.dy/dx + (e^u).d²y/dx².dx/du
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Updated: August 30, 2004
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