求由方程x^2+y^2=e^yarctgx确定的隐函数y=y(x)的微分dy.

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求由方程x^2+y^2=e^yarctgx确定的隐函数y=y(x)的微分dy.
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求由方程x^2+y^2=e^yarctgx确定的隐函数y=y(x)的微分dy.
求由方程x^2+y^2=e^yarctgx确定的隐函数y=y(x)的微分dy.

求由方程x^2+y^2=e^yarctgx确定的隐函数y=y(x)的微分dy.
两边对x求导得
2x+2yy'=e^y/(1+x²)+e^yarctgx
移项
2yy'=e^y/(1+x²)+e^yarctgx-2x
化简
y'=【e^y/(1+x²)+e^yarctgx-2x】/【2y】

dy/dx=【e^y/(1+x²)+e^yarctgx-2x】/【2y】

dy=【e^y/(1+x²)+e^yarctgx-2x】/【2y】dx

dx²+dy²=e^yd(arctany)+arctanyde^y
2xdx+2ydy=e^y/(1+y²) dy+arctany*e^ydy
所以dy=[e^y/(1+y²)+arctany*e^y-2y]/2x dx

2xdx+2ydy=e^ydyarctgx+e^ydx/(1+x^2),整理穿dy=【e^y/(1+x^2)-2x】dx/(2y-e^yarctgx)