[求助] 一阶微分方程求解 已有2人参与

3楼: Originally posted by aprillf at 2016-06-13 21:38:29
通解简单的形式应该是类似这样；

\sqrt{\frac{g}{A}}\tanh\left(\left({\color{red}C_1}+t\right)\sqrt{gA}\right)

tanh 表示双曲正切 tanh(x)=（e^x-e^-x）/(e^x+e^-x)

2楼: Originally posted by peterflyer at 2016-06-13 14:51:25
dv/dt=g-A*v^2=g*(1-A/g*v^2)
dv/(1-A/g*v^2)=g*dt
{1/+1/}*dv=2*g*dt
Ln{/}=2*g*sqrt(A/g)*t + LnC
由t=0 v=0代入上式得到C=1
/}=exp{2*sqrt(A*g)*t}
v=sqrt(g/A)*sh/ch...