Question

高等数学第二章习题 求详解

Answer 1

dx/dt=2
e^y+te^ydy/dt+dy/dt=0 dy/dt=-e^y/(1+te^y)
dy/dx=(dy/dt)/(dx/dt)=-e^y/[2(1+te^y)]
d[dy/dx)]/dt=[-2(1+te^y)e^ydy/dt+2e^y(e^y+te^ydy/dt)]/[4(1+te^y)^2]
=[2(1+te^y)e^y*e^y/(1+te^y)+2e^y(e^y-te^y*e^y/(1+te^y))]/[4(1+te^y)^2]
=[2e^(2y)+2e^(2y)(1-te^y/(1+te^y))]/[4(1+te^y)^2]
=[4e^(2y)-2te^(3y)/(1+te^y)]/[4(1+te^y)^2]
d^2(y)/dx^2=d[dy/dx)]/dt/(dx/dt)=[2e^(2y)-te^(3y)/(1+te^y)]/[4(1+te^y)^2]
t=0 y=-1
d^2(y)/dx^2|(t=0,y=-1)=e^(-2)/2=1/(2e^2) 选C