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    Help with differential equation assignment

    Consider the third order non-linear differential equation:

    y'y'''=y''

    This equation has two "obvious" families of solutions. What are they?

    I know that one family of solutions is C1(an arbituary constant), but I cant work out what the other family of solutions is. TIA for any help.
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    Quote Originally Posted by dgmossman View Post
    Consider the third order non-linear differential equation:

    y'y'''=y''

    This equation has two "obvious" families of solutions. What are they?

    I know that one family of solutions is C1(an arbituary constant), but I cant work out what the other family of solutions is. TIA for any help.
    The constant works. You also have that y'' = y''' = 0 when y = Ax, but then I would have thought that the first family would be y = Ax + B in that case.

    -Dan
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    Quote Originally Posted by dgmossman View Post
    Consider the third order non-linear differential equation:

    y'y'''=y''

    This equation has two "obvious" families of solutions. What are they?

    I know that one family of solutions is C1(an arbituary constant), but I cant work out what the other family of solutions is. TIA for any help.
    \frac{dy}{dx} \, \frac{d^3 y}{dx^3} = \frac{d^2 y}{dx^2}.

    First make the substitution u = \frac{dy}{dx}:

    u \frac{d^2 u}{dx^2} = \frac{du}{dx} \Rightarrow \frac{d^2 u}{dx^2} = \frac{1}{u} \, \frac{du}{dx}.

    Now make the substitution w = \frac{du}{dx}. Note that \frac{d^2 u}{dx^2} = \frac{dw}{dx} = \frac{dw}{du} \cdot \frac{du}{dx} = w \, \frac{dw}{du}:

    w \, \frac{dw}{du} = \frac{1}{u} \, w \Rightarrow w \left( \frac{dw}{du} - \frac{1}{u} \right) = 0.

    Case 1: w = 0 \Rightarrow \frac{du}{dx} = 0 \Rightarrow u = C \Rightarrow \frac{dy}{dx} = C \Rightarrow y = Cx + D.

    Case 2: \frac{dw}{du} - \frac{1}{u} = 0.
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