Question

求解如图所示的高阶齐次微分方程。

a,b,c,d,分别为4阶，3阶，2阶，1阶项前面的常数。

Answer 1

令p=y'，则ap'''+(b/x)*p''+(c/x^2)*p'+(d/x^3)*p=0
ax^3*p'''+bx^2*p''+cx*p'+d*p=0
令x=e^t，则t=lnx
p'=dp/dx=(dp/dt)*(dt/dx)=(1/x)*(dp/dt)

p''=(1/x^2)*(d^2p/dt^2-dp/dt)
p'''=(1/x^3)*(d^3p/dt^3-3d^2p/dt^2+2dp/dt)
所以a*(d^3p/dt^3-3d^2p/dt^2+2dp/dt)+b*(d^2p/dt^2-dp/dt)+c*dp/dt+d*p=0
a*d^3p/dt^3+(b-3a)*d^2p/dt^2+(2a-b+c)*dp/dt+d*p=0
特征方程为：ar^3+(b-3a)r^2+(2a-b+c)r+d=0

Answer 2

y=C