example of derivative as parameter

For solving the (nonlinear) differential equationMathworldPlanetmath

x=y3p-2py2 (1)

with  p=dydx,  according to III in the parent entry (http://planetmath.org/DerivativeAsParameterForSolvingDifferentialEquations), we differentiate both sides in regard to y, getting first


Removing the denominators, we obtain


The left hand side can be factored:

(ydpdy+2p)(1+6p2y)=0 (2)

Now we may use the zero rule of product; the first factor of the product in (2) yields  ydpdy=-2p, i.e.


whence  y2=Cp,  i.e.  p=Cy2.  Substituting this into the original equation (1) we get  x=y33C-2C.  Hence the general solution of (1) may be written


The second factor in (2) yields  6p2y=-1,  which is substituted into (1) multiplied by 3p:


Thus we see that  p=2y3x, which is again set into (1), giving


Finally, we can write it


which (a variant of the so-called semicubical parabola) is the singular solution of (1).

Title example of derivative as parameter
Canonical name ExampleOfDerivativeAsParameter
Date of creation 2013-03-22 18:29:03
Last modified on 2013-03-22 18:29:03
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Example
Classification msc 34A05
Synonym example of solving an ODE