Question

求微分方程xy''+y'=0的通解（提示：可降阶）

Answer 1

解法一：∵xy''+y'=0
==>xdy'/dx=-y'
==>dy'/y'=-dx/x
==>ln│y'│=-ln│x│+ln│C1│
(C1是积分常数)
==>y'=C1/x
==>y=C1ln│x│+C2
(C2是积分常数)
∴原方程的通解是y=C1ln│x│+C2
(C1,C2是积分常数)；
解法二：∵令t=ln│x│，则xy'=dy/dt，x²y''=d²y/dt²-dy/dt
代入原方程得
d²y/dt²-dy/dt+dy/dt=0
==>
d²y/dt²=0
==>dy/dt=C1
(C1是积分常数)
==>y=C1t+C2
(C2是积分常数)
==>y=C1ln│x│+C2
∴原方程的通解是y=C1ln│x│+C2
(C1,C2是积分常数)。