a) Solve the differential equation: day dx4 -3d²-4y = 12sin2x - 10e-²x
Answers
Answer:
Method of Undetermined Coefficients
Start with the homogeneous equation and the complementary solution :
y
'
'
−
4
y
'
+
4
y
=
0
This has characteristic equation:
λ
2
−
4
λ
+
4
=
0
⇒
(
λ
−
2
)
2
=
0
Repeated roots mean that, in lieu of the usual solution
y
c
=
α
e
λ
1
x
+
β
e
λ
2
x
, we look here for a solution in the form:
y
c
=
e
2
x
(
α
x
+
β
)
And because the non-homogeneous equation already has a
e
2
x
term, we must look at a particular solution in the form:
y
p
=
γ
x
2
e
2
x
⇒
y
'
=
2
γ
x
e
2
x
+
2
γ
x
2
e
2
x
⇒
y
'
'
=
2
γ
e
2
x
+
8
γ
x
e
2
x
+
4
γ
x
2
e
2
x
Putting these into the equation:
2
γ
e
2
x
+
8
γ
x
e
2
x
+
4
γ
x
2
e
2
x
−
4
(
2
γ
x
e
2
x
+
2
γ
x
2
e
2
x
)
+
4
γ
x
2
e
2
x
=
2
e
2
x
⇒
γ
=
1
and
y
p
=
x
2
e
2
x
The general solution is:
y
g
=
y
c
+
y
p
y
g
=
e
2
x
(
α
x
+
β
)
+
x
2
e
2
x
=
e
2
x
(
x
2
+
α
x
+
β
)
Now applying the IV's:
y
(
0
)
=
1
⇒
β
=
1
⇒
y
g
=
e
2
x
(
x
2
+
α
x
+
1
)
y
'
=
2
e
2
x
(
x
2
+
α
x
+
1
)
+
e
2
x
(
2
x
+
α
)
=
e
2
x
(
2
x
2
+
(
2
α
+
2
)
x
+
(
2
+
α
)
)
And from the second IV,
y
'
(
0
)
=
4
⇒
α
=
2
y
g
=
e
2
x
(
x
2
+
2
x
+
1
)