Question

常微分方程组求解

y^3=[(B/A)x^3]+[(C/D)z^3]+Ex=y=z=x0时，能求出E值。x=0时，y与z不等于0

Answer 1

①x'=A*[1/(y-x)]*(y/x)
2xx'=2Ay/(y-x)
(x^2)'=2Ay/(y-x)
(B/A)*(x^2)'=2By/(y-x)
②z'=D*[1/(z-y)]*(y/z)
2zz'=2Dy/(z-y)
(z^2)'=2Dy/(z-y)
(C/D)*(z^2)'=2Cy/(z-y)
③y'=B*[1/(y-x)]*(x/y)+C*[1/(z-y)]*(z/y)
2yy'=2Bx/(y-x)+2Cz/(z-y)
(y^2)'=2By/(y-x)-2B+2Cy/(z-y)+2C
③-①-②，(y^2)'-(B/A)*(x^2)'-(C/D)*(z^2)'=2C-2B
[y^2-(B/A)*x^2-(C/D)*z^2]'=2C-2B
y^2-(B/A)*x^2-(C/D)*z^2=(2C-2B)*t+E，其中E是任意常数
当t=0时，x=y=z=x0，则E=(1-B/A-C/D)*x0^2
所以y^2-(B/A)*x^2-(C/D)*z^2=(2C-2B)*t+(1-B/A-C/D)*x0^2
两边同乘以AD
ADy^2-BDx^2-ACz^2=(2ACD-2ABD)*t+(AD-BD-AC)*x0^2

追问

能求出x,y,z各自关于t的表达式吗

追答

求不出的，因为你这个方程组不是线性的