Question

Помогите решить дефференциальныое уравнение первого порядка.

Answer 1

$x^2y'=3x^2+xy-y^2\; |:x^2\ne 0\\\\y'=3+ \frac{y}{x} -(\frac{y}{x} )^2\\\\u= \frac{y}{x} \; ,\; \; y=ux\; ,\; \; y'=u'x+u\\\\u'x+u=3+u-u^2\\\\ \frac{du}{dx} \cdot x=3-u^2\\\\ \int \frac{du}{3-u^2}= \int \frac{dx}{x} \\\\ \frac{1}{2\sqrt3}\cdot ln\Big | \frac{\sqrt3+u}{\sqrt3-u} \Big |= ln |x|+ln|C|\\\\ln\Big | \frac{x\sqrt3+y}{x\sqrt3-y} \Big |=2\sqrt3\cdot ln|Cx|\\\\ \frac{x\sqrt3+y}{x\sqrt3-y} =(Cx)^{2\sqrt3}$