神马作文网谁知道这道微分方程的计算方式啊
来源：学生作业帮 编辑：神马作文网作业帮 分类：数学作业 时间：2024/11/19 23:59:38
谁知道这道微分方程的计算方式啊
d^2y/dx^2=a-(dy/dx)^2*(1/y).方程里的a是常数.
d^2y/dx^2=a-(dy/dx)^2*(1/y).方程里的a是常数.
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）
证毕
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）
证毕