y''+2y'+y=e^{-x},y\left(0\right)=0,y'\left(0\right)=1 ODE
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1
Answer:
Solve:
a.
y
′′
−
2
y
′
+
2
y
=
0
where
y
(
0
)
=
0
and
y
′
(
π
)
=
1
.
b.
y
′′
+
4
y
′
+
5
y
=
35
e
−
4
x
where
y
(
0
)
=
1
and
y
′
(
0
)
=
0
.
Laplace Transform as a Tool for Solving IVPs:
In this problem we shall utilize the usefulness of Laplace transform for solving initial value problems, in particular, we shall use the following properties
For a given function
F
(
t
)
L
{
F
′
(
t
)
}
=
s
L
F
(
t
)
−
F
(
0
)
L
{
F
′′
(
t
)
}
=
s
2
L
F
(
t
)
−
s
F
(
0
)
−
F
′
(
0
)
.
L
{
e
a
t
F
(
t
)
}
=
f
(
s
−
a
)
where
L
{
F
(
t
)
}
=
f
(
s
)
.
Moreover, we will use the following known transformation:
L
{
cos
(
k
t
)
}
=
s
s
2
+
k
2
L
{
sin
(
k
t
)
}
=
k
s
2
+
k
2
L
{
e
a
t
}
=
1
s
−
a
hope it helps you
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