一道大一上高数题.设f(x)是有二阶连续偏导数,f(0)=0,f'(0)=1.且积分∫L[xy(x+y)-f(x)y]dx+[f'(x)+x2y]dy与路径无关求f(x)∫L是线积分.x2y是x方乘以y即x^2*y

Question

一道大一上高数题.
设f(x)是有二阶连续偏导数,f(0)=0,f'(0)=1.且积分∫L[xy(x+y)-f(x)y]dx+[f'(x)+x2y]dy与路径无关 求f(x)
∫L是线积分. x2y是x方乘以y即x^2*y

Answer 1

[xy(x+y)-f(x)y]对y求偏导=x^2+2xy-f(x)
[f'(x)+x2y]对y求x偏导=f"(x)+2xy
利用与路径无关的条件：f"(x)+2xy=x^2+2xy-f(x)
求解微分方程：f(x)=sinx+2cosx+x^2-2