Question

X^2 *(y'-1)=2*y' дифф уравнение

Answer 1

X²*y'-2*y'=x², y'*(x²-2)=x², y'=dy/dx=x²/(x²-2), dy=x²*dx/(x²-2). Интегрируя, находим y=x+1/(2*√2)*ln/(x-√2)/(x+√2)/+C. Ответ: y=x+1/(2*√2)*ln/(x-√2)/(x+√2)/+C.

Answer 2

$x^2(y'-1)=2y'\\\\x^2y'-x^2=2y'\\\\y'(x^2-2)=x^2\\\\ \frac{dy}{dx}\cdot (x^2-2)=x^2\\\\\int dy=\int \frac{x^2\, dx}{x^2-2}\\\\\int dy=\int \frac{(x^2-2)+2}{x^2-2}\, dx\\\\\int dy=\int (1+\frac{2}{x^2-2})\, dx\\\\y=x+2\cdot \frac{1}{2\sqrt2}\cdot ln\Big |\frac{x-\sqrt2}{x+\sqrt2}\Big |+C$