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Differential Equations                                        Quiz #3                                                 March 11, 2005

Name____________________                   R.  Hammack                                                 Score ______
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(1)    Find the orthogonal trajectory of   y^3+3x^2y=c.

Differentiating implicitly with respect to x gives
3y^2y'+6x y+3x^2y'=0,
from which we get y'=-(2x y)/(x^2 + y^2)

Thus, the orthogonal trajectory must have D.E.
dy/dx=(x^2 + y^2)/(2x y) or   2x y dy =(x^2 + y^2)dx

This is a homogeneous differential equation.
We use the substitution y = ux,  dy = x du +u dx.

  2x^2 u (x du + u dx) = (x^2 + x^2u^2) dx  2x^3 u  du + 2x^2 u^2dx = x^2 dx ... c x(1 - (y/x)^2)  1 = c (x - y^2/x)  x = c (x^2 - y^2)  a x = x^2 - y^2