Question

The general solution of the DE ` x ^(2) (dy)/(dx) = x ^(2) + xy + y ^(2) ` is
A. ` tan ^(-1) ""(y)/(x) = log x + C `
B. ` tan ^(-1)""(x)/(y) =log x + C `
C. ` tan ^(-1) ""(y)/(x)=log y +C `
D. none of these

Answer 1

Correct Answer - A
` (dy)/(dx) = d ( x ^(2) + xy + y ^(2))/( x ^(2))`, which is homogeneous.
Put `y = vx and (dy)/(dx) = v + ( dv)/(dx) ` to get ` v + x (dv)/(dx) = (1 + v + v ^(2))`
` therefore int (dv)/((1 + v ^(2))) = int (1)/(x) dx rArr tan ^(-1) v = log x + C rArr tan ^(-1) ""(y)/(x) = log x +C`.