Kendra F. answered • 08/04/17

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The differentiable equation is not separable, so use the variable substitution; v = y/x

Switch it around to: y = vx ; substitute vx in for y

Derivative of y = vx gets subbed in for dy/dx

dy/dx = (dv/dx)x + v

dy/dx = (r√(x

^{2}+y^{2}) + by) / bx(dv/dx)x + v = r(x

^{2}+(vx)^{2})^{1/2}/bx +bvx/bx(dv/dx)x + v = r(x

^{2}+v^{2}x^{2})^{1/2}/bx + v(dv/dx)x = rx(v

^{2}+1)^{1/2 }/bx(dv/dx)x = (r/b)(v

^{2}+1)^{1/2 }It is now separable

(v

^{2}+1)^{-1/2 }dv = (r/b)(1/x) dxIntegrate

∫ (v

^{2}+1)^{-1/2 }dv = r/b ∫ 1/x dxln|v+(v

^{2}+1)^{1/2}| + c = (r/b)ln|x| + cconsolidate constants ; sub in y/x for v

ln|y/x + (y

^{2}+ x^{2})^{1/2 }/x| = (r/b)ln|x| + ctake it to the power of e

y/x + (y

^{2}+ x^{2})^{1/2 }/x = e^{c}x^{r/b }e

^{c}is just another constant; so is r/b which looks to be k in the solution.multiply both sides by x

y + (y

^{2}+ x^{2})^{1/2 }= C*x*x^{k }y + (y

^{2}+ x^{2})^{1/2 }= Cx^{(k+1) }I think there should be a constant multiplied by x

^{(k+1) }but otherwise appears to be correct. I hope this helps.

Andy C.

08/05/17