帮忙解一个微分方程:f^2(x) =2xf(x)/3 +x^2 f'(x)/3,且f(2)=2/9 ,求y=f(x)的表达式.
问题描述：
帮忙解一个微分方程:f^2(x) =2xf(x)/3 +x^2 f'(x)/3,且f(2)=2/9 ,求y=f(x)的表达式.
f^2(x)是f（x）的平方,f'(x)是f(x)的导数.
答
令y=f(x)y^2=2xy/3+x^2y'/33y^2=2xy+x^2y'3(y/x)^2=2(y/x)+y'令u=y/x y=ux y'=u'x+u3u^2=3u+u'x3u^2-3u=xdu/dxdx/x=du/(3u^2-3u)∫dx/x=∫du/3u(u-1)ln|x|=1/3*(ln|u-1|-ln|u|)+Cx^3=C*(u-1)/u因为f(2)=2/9所以当x=2...