对原式两边微分
yx^(y-1)dx+x^ylnxdy=y^xlnydx+xy^(x-1)dy
(x^ylnx-xy^(x-1))dy=(y^xlny-yx^(y-1))dx
两边除以(x^ylnx-xy^(x-1))dx得
y'=dy/dx=(y^xlny-yx^(y-1))/(x^ylnx-xy^(x-1))
对上面第2式两边微分
d(x^ylnx-xy^(x-1))dy+(x^ylnx-xy^(x-1))ddy=d(y^xlny-yx^(y-1))dx
(yx^(y-1)lnxdx+x^y(lnx)^2dy+x^y/xdx-y^(x-1)dx-xy^(x-1)lnydx-x(x-1)y^(x-2)dy)dy+(x^ylnx-xy^(x-1))ddy
=(y^x(lny)^2dx+xy^(x-1)lnydy+y^x1/ydy-x^(y-1)dy-y(y-1)x^(y-2)dx-yx^(y-1)lnxdy)dx
两边除以dx^2
(yx^(y-1)lnx+x^y(lnx)^2y'+x^y/x-y^(x-1)-xy^(x-1)lny-x(x-1)y^(x-2)y')y'+(x^ylnx-xy^(x-1))y''
=(y^x(lny)^2+xy^(x-1)lnyy'+y^x1/yy'-x^(y-1)y'-y(y-1)x^(y-2)-yx^(y-1)lnxy')
得2阶导数
y''=((y^x(lny)^2+xy^(x-1)lnyy'+y^x1/yy'-x^(y-1)y'-y(y-1)x^(y-2)-yx^(y-1)lnxy')-((yx^(y-1)lnx+x^y(lnx)^2y'+x^y/x-y^(x-1)-xy^(x-1)lny-x(x-1)y^(x-2)y')y'))/(x^ylnx-xy^(x-1))
晕死了，默哀5分钟
要是再把上面的第3式一阶导数y'代进去
就会得到史上
最颠三倒四
最啰嗦
最花里胡哨
最长的x、y字符串
装进骨灰盒了
不用再默哀
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