Question

设y=y(x)由方程组x=3t^2+2t+3，e^ysint-y+1=0所确定，求当t=0时，d^

设y=y(x)由方程组x=3t^2+2t+3，e^ysint-y+1=0所确定，求当t=0时，d^2y/dx^2

Answer 1

解：当t=0时，x=3 y=1
dx/dt=6t+2
e^ysintdy+e^ycostdt-dy=0,dy/dt=e^ycost/(1-e^ysint)
dy/dx=e^ycost/[(1-e^ysint)(6t+2)]
d^2y/dx^2=d(dy/dx)/dt*dt/dx=d{e^ycost/[(1-e^ysint)(6t+2)]}*1/(6t+2)
={(-e^tsint)(1-e^ysint)(6t+2)-e^ycost*[6(1-e^ysint)-e^ycost(6t+2)]}/[(1-e^ysint)^2*(6t+2)^3]
将t=0,x=3,y=1带入上式，有
d^2y/dx^2=(e^2-3e)/4