题目
2)y'=(x)/(y)+(y)/(x),y|_(x=1)=2;
2)$y'=\frac{x}{y}+\frac{y}{x},y|_{x=1}=2;$
题目解答
答案
令 $ u = \frac{y}{x} $，则 $ y = ux $，代入原方程得
\[
u + x \frac{du}{dx} = \frac{1}{u} + u \implies x \frac{du}{dx} = \frac{1}{u} \implies u \, du = \frac{1}{x} \, dx.
\]
积分得
\[
\frac{u^2}{2} = \ln |x| + C \implies \frac{y^2}{2x^2} = \ln |x| + C \implies y^2 = 2x^2 (\ln |x| + C).
\]
由 $ y|_{x=1} = 2 $，得 $ C = 2 $，故特解为
\[
\boxed{y^2 = 2x^2 (\ln x + 2)}.
\]