Question

常微分方程的通解

dy/dx=(x-y+1)/(x+y-3)
y^4=2y^n+y=0
y''+6y'+9y=e^(-3x)
y''+y'-2y=4e^(2x)

Answer 1

先求齐次通解，再求非齐次通解，然后组合就行了。

Answer 2

1
dy/dx=(x-y+1)/(x+y-3)
设u=x-y+1
v=x+y-3
x=(u+v)/2-1
y=(v-u)/2-2
dx=(du+dv)/2
dy=(dv-du)/2
(dv-du)/(du+dv)=u/v
udu+udv=vdv-vdu
udu+udv+vdu-vdv=0
u^2+2uv-v^2=C0
通解
(x-y+1)^2+2(x-y+1)(x+y-3)-(x+y-3)^2=C0

y''+6y'+9y=e^(-3x)
y''+6y'+9y=0
特征方程
r^2+6r+9=0
r=-3
y=C1e^(-3x)+C2xe^(-3x)
设y=C(x)e^(-3x)
C''+6C'=1
dC'/dx=1-6C'
-6dC'/(1-6C')=-6dx
ln(1-6C')=-6x+C2
1-6C'=C3*e^(-6x)
C'=1/6-C3e^(-6x)/6
C=x/6+C3e^(-6x)/36
y=[x/6+C3e^(-6x)/36]e^(-3x)+C1e^(-3x)+C2xe^(-3x)

Answer 3