Help with differential equation assignment

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August 8th 2008, 03:31 PM
dgmossman

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.
August 8th 2008, 07:20 PM
topsquark

Quote:

Originally Posted by dgmossman

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
August 8th 2008, 08:28 PM
mr fantastic

Quote:

Originally Posted by dgmossman

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