高等数学微分方程d^2y/dx^2=a-(dy/dx)^2*(1/y).方程里的a是常数.请在下面的对话框输入此微分方程的计算公式:(X^2就是X的2次方)
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高等数学微分方程d^2y/dx^2=a-(dy/dx)^2*(1/y).方程里的a是常数.请在下面的对话框输入此微分方程的计算公式:(X^2就是X的2次方)
高等数学微分方程
d^2y/dx^2=a-(dy/dx)^2*(1/y).方程里的a是常数.
请在下面的对话框输入此微分方程的计算公式:
(X^2就是X的2次方)
高等数学微分方程d^2y/dx^2=a-(dy/dx)^2*(1/y).方程里的a是常数.请在下面的对话框输入此微分方程的计算公式:(X^2就是X的2次方)
首先,假设你已经知道啥叫微分方程.
一般的微分方程是没办法直接解出精确的解来的.
但是我们大多数情况下遇到的方程是可以有现成的解法的.具体这里不讲了.你只要随便去弄本讲微分方程的书看看就懂了.
当然你事先要好好学下数学分析.这里推荐《微积分学教程》(菲赫金戈尔兹著,九章数学书店有售)
其实通常情况下,我们并不是直接求方程的精确解,而是把它大致上变成一个差分方程来求近似解.我曾经给人详细讲过的:
对于本题目,
令y=z^(1/2)
则dy/dx=dy/dz*dz/dx=1/2*z^(-1/2)*dz/dx -----(1)
又d(z^(-1/2))/dx=*(-1/2)*z^(-3/2)*dz/dx
d^2y/dx^2=d(dy/dx)/dx
=1/2*z^(-1/2)*d^2z/dx^2+1/2*(-1/2)*z^(-3/2)*dz/dx*dz/dx
=1/2*z^(-1/2)*d^2z/dx^2+1/2*(-1/2)*z^(-3/2)*(dz/dx)^2 --(2)
将(1)(2)代入原微分方程
1/2*z^(-1/2)*d^2z/dx^2-1/4*z^(-3/2)*(dz/dx)^2
=a-(1/2*z^(-1/2)*dz/dx)^2/ z^(1/2)
=a-1/4*z^(-3/2)*(dz/dx)^2
1/2*z^(-1/2)*d^2z/dx^2=a =>d^2z/dx^2=2a*z^(1/2) ----(3)
令dz/dx=p =>d^2z/dx^2=dp/dx=dp/dz*(dz/dx)=dp/dz*p
代入(3)有dp/dz*p=2a*z^(1/2) =>p*dp=2a*z^(1/2)*dz
两边分别积分=>p^2=a*4/3z(3/2)+C
=>p=dz/dx=a*4/3z(3/2)+C =>dz/(a*4/3*z(3/2)+C)=dx ----(4)
为书写方便令D=(C/(4a/3))^(1/3) 令3/(4a)=b
(4)b/(z(3/2)+D^3)=dx ,又z(1/2)=y带入后可得dx/dy=2bt/(y^3+D^3)
=>x+C=2b/(3D)*(ln(y+D)+1/α*(ln(y+D*α)+1/β*(ln(y+D*β)))
其中,1,α,β为1的三次单位根,即x^3-1=(x-1)(x-α)(x-β)
令y=z^(1/2)
则dy/dx=dy/dz*dz/dx=1/2*z^(-1/2)*dz/dx -----(1)
又d(z^(-1/2))/dx=*(-1/2)*z^(-3/2)*dz/dx
d^2y/dx^2=d(dy/dx)/dx
=1/2*z^(-1/2)*d^2z/dx^2+1/2*(-1/2)*z^(-...
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令y=z^(1/2)
则dy/dx=dy/dz*dz/dx=1/2*z^(-1/2)*dz/dx -----(1)
又d(z^(-1/2))/dx=*(-1/2)*z^(-3/2)*dz/dx
d^2y/dx^2=d(dy/dx)/dx
=1/2*z^(-1/2)*d^2z/dx^2+1/2*(-1/2)*z^(-3/2)*dz/dx*dz/dx
=1/2*z^(-1/2)*d^2z/dx^2+1/2*(-1/2)*z^(-3/2)*(dz/dx)^2 --(2)
将(1)(2)代入原微分方程
<=>1/2*z^(-1/2)*d^2z/dx^2-1/4*z^(-3/2)*(dz/dx)^2
=a-(1/2*z^(-1/2)*dz/dx)^2/ z^(1/2)
=a-1/4*z^(-3/2)*(dz/dx)^2
<=>1/2*z^(-1/2)*d^2z/dx^2=a =>d^2z/dx^2=2a*z^(1/2) ------(3)
令dz/dx=p =>d^2z/dx^2=dp/dx=dp/dz*(dz/dx)=dp/dz*p
代入(3)有dp/dz*p=2a*z^(1/2) =>p*dp=2a*z^(1/2)*dz
两边分别积分=>p^2=a*4/3z(3/2)+C
=>p=dz/dx=a*4/3z(3/2)+C =>dz/(a*4/3*z(3/2)+C)=dx ------(4)
为书写方便令D=(C/(4a/3))^(1/3) 令3/(4a)=b
(4)<=>b/(z(3/2)+D^3)=dx ,又z(1/2)=y带入后可得dx/dy=2bt/(y^3+D^3)
=>x+C=2b/(3D)*(ln(y+D)+1/α*(ln(y+D*α)+1/β*(ln(y+D*β)))
其中,1,α,β为1的三次单位根,即x^3-1=(x-1)(x-α)(x-β)
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