二阶变系数线性微分方程化为标准型的求解

(2-10)
令 c1′ ( x) y1′ + c2′ ( x) y2′ + + cn′ ( x) yn′ =0
则
= yp′′ c1 ( x) y1′′ + c2 ( x) y2′′ + + cn ( x) yn′′
(2-11)
即 yp′′ 也具有 c1 ( x), c2 ( x),, cn ( x) 为常数时所具有的形式。
2. 标准型
二阶变系数线性微分方程
d2 y dx2
+
p1
(
x)
dy dx
+p2( Nhomakorabeax)
y
= Φ ( x)
(1-1)
其中 p1 ( x), p2 ( x), Φ ( x) 均为 x 的连续函数。
设
y = uv
(1-2)
其= 中 u u= ( x), v v ( x)
= dy u dv + v du dx dx dx
由(1-7)式可解得
u
=
−
e
1 2
∫
p1( x)dx
再由(1-7)~(1-9)式可得
2
d2u dx2
+
p1
(
x)
du dx
+
u
dp1 ( x)
dx
= 0
2
d2u dx2
−
1 2
p12
(x)u
+
u
dp1 ( x)
dx
= 0
d2u dx2
−
1 4
p12
(
x)u
+
1 2
u
dp1 ( x)
dx
= 0
= ddx2u2
( ) dn y
dn−1 y
dn−2 y
dy
dxn
+
p1
dxn−1
+
p2
dxn−2
++
pn−1
dx
+
pn y
=Φ
x
(2-4)
设其通解为
y ( x=) yc + yp
(2-5)
其中，对于
yc =
c1
y1
+
c2
y2
++
cn
yn
(2-6)
的求法(齐次方程的通解)——特征方程，特征根。
假设已知 yc ，现在求 yp 。
Solution of Two Order Variable Coefficient Linear Differential Equation into Standard Form
Xiong Chen1,2, Shiyou Lin1*, Haohan Zhang1 1School of Mathematics and Statistics, Hainan Normal University, Haikou Hainan 2Yilong County No.2 Middle School, Hainan Normal University, Sichuan Nanchong
(2-13)
= 其中 k 0,1, 2,, n − 2 。
( ) ( ) ( ) = y(pn−1)
c1
x y1(n−1) + c2
x
y2( n −1)
++
cn
x
yn( n −1)
y(pn)
=
c1 ( x)
y1(n)
+ c2
(x)
y2(n)
+ + cn
(x)
yn(n)
+
c1′
(x)
y1( n −1)
同理可求得 yp 的三阶导数为
( ) ( ) ( ) = yp′′′ c1 x y1′′′ + c2 x y2′′′ + + cn x yn′′′
(2-12)
依次求导至 n −1 阶的导数，且每一次的导数满足
c1′ ( x) y1(k) + c2′ ( x) y2(k) + + cn′ ( x) yn(k) =0
则
= y p′
c1
(
x)
y1′
+
c2
(
x)
y2′
+
+
cn
(
x)
yn′
(2-9)
即 yp′ 具有 c1 ( x), c2 ( x),, cn ( x) 为常数时所具有的形式。
对 yp 继续求二阶导数
= yp′′ c1 ( x) y1′′ + c2 ( x) y2′′ + + cn ( x) yn′′ + c1′ ( x) y1′ + c2′ ( x) y2′ + + cn′ ( x) yn′
Keywords
Two Order Variable Coefficient, Linear Differential Equation, Standard Type, Cofunction, Particular Integral, General Solution
二阶变系数线性微分方程化为标准型的求解
+ c2′
(x)
y2( n −1)
+ + cn′
(x)
yn( n −1)

(2-14) (2-15)
所以
= yp′ c1 ( x) y1′ + c2 ( x) y2′ + + cn ( x) yn′
= yp′′ c1 ( x) y1′′ + c2 ( x) y2′′ + + cn ( x) yn′′
时所具有的形式。
90
陈雄 等
所以，对(2-7)式求导可得
= yp′ c1 ( x) y1′ + c2 ( x) y2′ + + cn ( x) yn′ + c1′ ( x) y1 + c2′ ( x) y2 + + cn′ ( x) yn
(2-8)
令 c1′ ( x) y1 + c2′ ( x) y2 + + cn′ ( x) yn =0
( ) ( ) ( ) = yp′′′
c1
x y1′′′ + c2
x
y2′′′
++
cn
x
yn′′′

( ) ( ) ( ) = y(pn−1)
c1
x
y1(n−1) + c2
x
y2( n −1)
++
cn
x
yn( n −1)
= y(pn)
c1
(x)
y1(n)
+
c2
(
x)
y2(n)
++
cn
(x)
88
陈雄 等
u
d2v dx2
+
2
du dx
+
p1
( x)u
dv dx
+
d2u

dx
2
+
p1
(x)
du dx
+
p2
(
x
)u


v
= Φ ( x)
令 dv 的系数等于 0，即 dx
2
du dx
+
p1
(
x)u
= 0
du dx
=
−1 2
p1 ( x)u
2 du dx
=
− p1 ( x)u
Abstract
This paper discusses the solution of the two order variable coefficient linear differential equation with standard type, which transforms the traditional method of reducing order. Through simplifying the original differential equation and using means of cofunction and particular integral, we can get the homogeneous and non-homogeneous solution of the standard type. Finally we can construct the general solution of the original equation.
+

p2
(
x)
−
1 2
dp1 ( x)
dx
−
1 4
p12
( x) uv

= Φ ( x)
上式两边同时除以 u 后得
d2v dx2
+

p2
(x)
−
1 2
dp1 ( x)
dx
−
1 4
p12
( x) v

= Φ ( x) = Φ ( x)e
1 2
∫
u
p1( x)dx
即
d2v dx2
设待定函数
c1
(
x
)
,
c2
(
x
)
,
,
cn
(
x
)
，满足