高数导数二阶导数求答案~~好难哦

Question

高数导数二阶导数
求答案~~
好难哦

Answer 1

sin(xy) = ln(x+e) - lny + 1
当 x = 0,
0 = lne - lny + 1
lny = 2,y(0) = e²
cos(xy)[y + xy'] = 1/(x+e) - y'/y
∵ y(0) = e²
∴ e² = 1/e - y'(0)/e²
y'(0) = e(1 - e³)
-sin(xy)[y+xy']² + cos(xy)[2y'+xy''] = -1/(x+e)² - y''/y + (y')²/y²
∵ y(0) = e²,y'(0) = e(1 - e³)
∴ 2e(1 - e³) = -1/e² - y''(0)/e² + (1 - e³)²/e²
∴ 2e³(1 - e³) = -1 - y''(0) + (1 - e³)²
y''(0) = (1 - e³)² - 2e³(1 - e³) - 1
= 1 - 2e³ + e^6 - 2e³ + 2e^6 - 1
= -4e³ + 3e^6
= e³(3e³ - 4)