大学常微分dy/dx=sin(y)

网友回答

dy/dx = siny
dy/siny = dx
∫dy/siny = ∫dx
∫sinydy/sin²y = x + C
-∫dcosy/(1-cos²y) = x + C
-∫dcosy/(1-cosy)(1+cosy) = x + C
-½∫[1/(1-cosy) + 1(1+cosy)]dcosy = x + C
-∫[1/(1-cosy) + 1(1+cosy)]dcosy = 2x + C
-[-ln|1-cosy| + 1n|1+cosy|] = 2x + C
ln|1-cosy| - 1n|1+cosy|] = 2x + C
ln|(1-cosy)/(1+cosy)| = 2x + C
x = ½ln|(1-cosy)/(1+cosy)| + C
= ½ln|2sin²(y/2)/2cos²(y/2)| + C
= ln|tan(y/2)| + C
验证：两边对x求导,得：
1 = [1/tan（y/2)][sec²(y/2)][1/2]dy/dx
1 = [1/2sin(y/2)cos(y/2)]dy/dx
1 = [1/siny]dy/dx
dy/dx = siny
上面结果可以作为答案,也可以：
ln|tan(y/2)| = x + c
tan(y/2) = Ce^x
y = 2arctan(Ce^x)
======以下答案可供参考======
供参考答案1:
分离变量啊dy/siny=dx
cscydy=dx
ln|cscy-coty|=x+C
供参考答案2:
多麼熟悉啊。。可惜过了两三年了，早忘了。