设方程[A(y)]²+xsin[πA(y)]+2x-3=0,确定y是x的隐函数,A(y)可微且A(0)=A'(
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设方程[A(y)]²+xsin[πA(y)]+2x-3=0,确定y是x的隐函数,A(y)可微且A(0)=A'(0)=1,求dy/dx|y=0
![设方程[A(y)]²+xsin[πA(y)]+2x-3=0,确定y是x的隐函数,A(y)可微且A(0)=A'(](/uploads/image/z/19751341-13-1.jpg?t=%E8%AE%BE%E6%96%B9%E7%A8%8B%5BA%28y%29%5D%26sup2%3B%2Bxsin%5B%CF%80A%28y%29%5D%2B2x-3%3D0%2C%E7%A1%AE%E5%AE%9Ay%E6%98%AFx%E7%9A%84%E9%9A%90%E5%87%BD%E6%95%B0%2CA%28y%29%E5%8F%AF%E5%BE%AE%E4%B8%94A%280%29%3DA%27%28)
∵[A(y)]²+xsin[πA(y)]+2x-3=0,且A(y)可微
==>2A(y)A'(y)y'+sin[πA(y)]+πxA'(y)y'cos[πA(y)]+2=0 (等式两端对x求导数)
==>[2A(y)+πxcos[πA(y)]]A'(y)y'=-2-sin[πA(y)]
∴y'=-[2+sin[πA(y)]/{[2A(y)+πxcos[πA(y)]]A'(y)}
∵A(0)=A'(0)=1
∴dy/dx|(y=0)=y'|(y=0)
=-[2+sin[πA(0)]/{[2A(0)+πxcos[πA(0)]]A'(0)}
=-[2+sin(π)/[2+πxcos(π)]
=-2/(2-πx)
=2/(πx-2).
==>2A(y)A'(y)y'+sin[πA(y)]+πxA'(y)y'cos[πA(y)]+2=0 (等式两端对x求导数)
==>[2A(y)+πxcos[πA(y)]]A'(y)y'=-2-sin[πA(y)]
∴y'=-[2+sin[πA(y)]/{[2A(y)+πxcos[πA(y)]]A'(y)}
∵A(0)=A'(0)=1
∴dy/dx|(y=0)=y'|(y=0)
=-[2+sin[πA(0)]/{[2A(0)+πxcos[πA(0)]]A'(0)}
=-[2+sin(π)/[2+πxcos(π)]
=-2/(2-πx)
=2/(πx-2).
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