PLEASE HELP: Some equations which do not appear to be separable can be made so by means of a suitable substitution. By means of the substitution z=y⁄x solve the following equation: dy/dx=y^2/x^2 +y/x+1. ??

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Answer 1 · 2018-05-28T12:40:01

#y = xtan(ln absx+c)#

Let:

#z = y/x#

#y = xz#

#dy/dx = z+xdz/dx#

Substitute in the original equation:

#dy/dx = y^2/x^2+y/x+1#

to have:

#z+xdz/dx = z^2+z+1#

#xdz/dx = z^2+1#

which is now separable:

#dz/(z^2+1) = dx/x#

#arctanz = lnabsx +c#

#z = tan(ln abs x +c)#

and undoing the substitution:

#y = xtan(ln absx+c)#