MATH 315 FALL 1998 TEST 2 SOLUTION KEY
1.
(15 pts.) Consider the differential equation
y
'' +2
y
' +
y
= 0.
(a)
Find the general solution for the equation.
Answer:
Characteristic polynomial is
r
2
+2
r
+1 = (
r
+1)
2
, with roots 1, 1.
The general solution is therefore
c
1
e

t
+
c
2
te

t
.
(b)
Determine the Wronskian for fundamental solution set.
Answer:
2.
(15 pts.) Find the general solution for the differential equation
y
''' + 2
y
'' 
y
'  2
y
= 0.
Answer:
Characteristic polynomial is
r
3
+2
r
2

r
2 = (
r
1)(
r
+1)(
r
+2), with roots 1, 1, 2.
The general solution is therefore
c
1
e
t
+
c
2
e

t
+
c
3
e
2
t
.
3.
(15 pts.) Find the general solution for the differential equation
y
'''' + 3
y
''  4
y
= 4
e

t
,
assuming that the general solution to the homogeneous equation is given by
Answer:
A particular solution should have the form
Y
(
t
) =
Ate

t
.
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