Question
1.
I
16y" Yy'
+
-
=
ext
' =
Xe
y
y
y
=
0
*t
" 12ext
=
*+
16x2e
8xe
-
ext((bx"
+
1)
+
8x
-
e*t
=
=
0
0
8
=
X
**
second order
Eg.
Homgenous
1 3
y" + my
.
Sy
-
=
0
(e St)"
-
Equation
Characteristic
2
v + yv
(r + s)(v
v=
-
-
-
SeSt
+
y(e St))
-
-
Zoe St
_
-
Set 0
=
Se-St
=
o
5 =0
1) = 0
-St
4 ,e
5 1
,
-
Get
a
geneVI
Solution
1 1
.
lby" Gy
-
16r2
+
fr + 1
-
y
0
=
=
(4 1)(4r 1)
-
characteristic
02
=
0
-
Stwice)
v
=
↳
eitteat
(x(tet)" f(te +)
*
-
↑
Liste + etete)
1 4
.
y" by + ay
=
v2
0
-
-
br + 9
=
-
G(e
*+
-
+
text
Gte*t)
(x + 1)e3Y
+
=
0
+ent
solution :
solve for
educated
Yp(X)
(v-b)
guess
Ax3e3 + Bx 23x
+
=
b/AX3e3x
+
6Axe3Y + 9Ax&23x +9Ax ex 9AxeBX + 2Be3" 6Bxe3x
+
(AxPe3" Bxe3x)"
+
y(x)
y(x)
=
=
=
c,
e3x
+
Gxe
3X
"
+
+
EGB-12B /
GA +
yu(x) + yp(x)
c , e3" +(zxe3x + =x2 ex + tx
-
*
e3x
B
=
=
2B
=
/
=
A
=
Bx 2
+
6Bxe3x +9Bx2e3x
(xe3x terms)
(e3x term)
t
e3x))
9Ax3e3X +
-
-
9Bx
e3X
=
(x + 1)e3x
*
b)3Ay2e3x 3Ax3e3x 2Bxe3x+ 3Bx2e3x) + 9AX e3x + 9Bx2e3x
+
+
=
(x + 1) e3