关於常微分的计算将某个模型建模以后得到:dx(t)/dt = 57500 - 2.75*10^-7*exp(x(t)/11) - 76333*(sin(2*pi*60*t))^2x(0) = 250请教各路高手帮我解出x(t),如果有详细解法更加感激也会给予多的报酬,谢谢,还是谢谢!
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![关於常微分的计算将某个模型建模以后得到:dx(t)/dt = 57500 - 2.75*10^-7*exp(x(t)/11) - 76333*(sin(2*pi*60*t))^2x(0) = 250请教各路高手帮我解出x(t),如果有详细解法更加感激也会给予多的报酬,谢谢,还是谢谢!](/uploads/image/z/11455925-5-5.jpg?t=%E5%85%B3%E6%96%BC%E5%B8%B8%E5%BE%AE%E5%88%86%E7%9A%84%E8%AE%A1%E7%AE%97%E5%B0%86%E6%9F%90%E4%B8%AA%E6%A8%A1%E5%9E%8B%E5%BB%BA%E6%A8%A1%E4%BB%A5%E5%90%8E%E5%BE%97%E5%88%B0%3Adx%28t%29%2Fdt+%3D+57500+-+2.75%2A10%5E-7%2Aexp%28x%28t%29%2F11%29+-+76333%2A%28sin%282%2Api%2A60%2At%29%29%5E2x%280%29+%3D+250%E8%AF%B7%E6%95%99%E5%90%84%E8%B7%AF%E9%AB%98%E6%89%8B%E5%B8%AE%E6%88%91%E8%A7%A3%E5%87%BAx%28t%29%2C%E5%A6%82%E6%9E%9C%E6%9C%89%E8%AF%A6%E7%BB%86%E8%A7%A3%E6%B3%95%E6%9B%B4%E5%8A%A0%E6%84%9F%E6%BF%80%E4%B9%9F%E4%BC%9A%E7%BB%99%E4%BA%88%E5%A4%9A%E7%9A%84%E6%8A%A5%E9%85%AC%2C%E8%B0%A2%E8%B0%A2%2C%E8%BF%98%E6%98%AF%E8%B0%A2%E8%B0%A2%21)
关於常微分的计算将某个模型建模以后得到:dx(t)/dt = 57500 - 2.75*10^-7*exp(x(t)/11) - 76333*(sin(2*pi*60*t))^2x(0) = 250请教各路高手帮我解出x(t),如果有详细解法更加感激也会给予多的报酬,谢谢,还是谢谢!
关於常微分的计算
将某个模型建模以后得到:
dx(t)/dt = 57500 - 2.75*10^-7*exp(x(t)/11) - 76333*(sin(2*pi*60*t))^2
x(0) = 250
请教各路高手帮我解出x(t),如果有详细解法更加感激也会给予多的报酬,谢谢,还是谢谢!
这个方程式经过一下修改,原本:
dy/dx = 57500 - 2.75*10^-7*exp(y/11) - 76333*(sin(2*pi*60*x))^2
先将复杂的系数设为a1~a3,以减轻复杂度:
dy/dx = a1 - a2*exp(cy) - a3*(sin(x))^2, a1~a3,c都是已知常数。
接著设 z = exp(cy) => y = ln(z)/c, 代回原式:
d(ln(z)/c)/dx = a1 - a2*z - a3*(sinx)^2
=> (1/cz)dz/dx = a1 - a2*z - a3*(sinx)^2
=> dz/dx = a1*cz - a2*cz^2 - a3*(sinx)^2*cz, c*a1~a3等都是常数,简化为 :
dz/dx = c1*z - c2*z^2 - c3*z*(sinx)^2 , c1~c3都是已知常数,且z(0)=exp(cy(0))也是已知常数
求z函数的解 ,希望能列出详细算式,感激不尽!
关於常微分的计算将某个模型建模以后得到:dx(t)/dt = 57500 - 2.75*10^-7*exp(x(t)/11) - 76333*(sin(2*pi*60*t))^2x(0) = 250请教各路高手帮我解出x(t),如果有详细解法更加感激也会给予多的报酬,谢谢,还是谢谢!
dy/dx = a1 - a2*exp(cy) - a3*(sin(x))^2,a1~a3,c都是已知常数.
先求1个特解,
dt/dx = a1 - a3*(sinx)^2 = a1 - a3[1 - cos(2x)]/2 = a1 - a3/2 + a3cos(2x)/2.
t = (a1 - a3/2)x + a3sin(2x)/4
再设 u = y - t,y = u + t,
du/dx = dy/dx - dt/dx = a1 - a2*exp(cy) - a3*(sin(x))^2 - [a1 - a3(sinx)^2] = -a2*exp{c[u + (a1 - a3/2)x + a3sin(2x)/4)]}
du/dx = -a2*exp{cu}*exp{c[(a1 - a3/2)x + a3sin(2x)/4)]},
-exp{-cu}du = a2*exp{c[(a1 - a3/2)x + a3sin(2x)/4)]}dx
d[exp(-cu)] = ca2*exp{c[(a1 - a3/2)x + a3sin(2x)/4)]}dx,
exp(-cu) = S ca2*exp{c[(a1 - a3/2)x + a3sin(2x)/4)]}dx 【求不定积分】
记F(x) = S ca2*exp{c[(a1 - a3/2)x + a3sin(2x)/4)]}dx
【F(x)为ca2*exp{c[(a1 - a3/2)x + a3sin(2x)/4)]}的原函数】
则,
-cu = ln|F(x)|,
u = (-1/c)ln|F(x)|.
y = u + t = (a1 - a3/2)x + a3sin(2x)/4 - 1/cln|F(x)|.
可是,
给出F(x)的解析表达式超出了俺的能力,俺只能到这了.