Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Mathematics
Mathematics
Problem
Variables
x
x
x
and
y
y
y
satisfy the differential equation
d
y
d
x
=
x
+
2
y
\frac{dy}{dx} = x + 2y
d
x
d
y
=
x
+
2
y
, and
y
(
0
)
=
A
y(0) = A
y
(
0
)
=
A
. Write
y
=
∑
k
=
0
∞
a
k
x
k
y = \sum_{k=0}^{\infty} a_k x^k
y
=
∑
k
=
0
∞
a
k
x
k
. a. Express
a
1
a_1
a
1
and
a
2
a_2
a
2
in terms of
A
A
A
and show that, for
k
≥
2
k \ge 2
k
≥
2
,
a
k
+
1
=
2
k
+
1
a
k
a_{k+1} = \frac{2}{k+1} a_k
a
k
+
1
=
k
+
1
2
a
k
. b. Hence find the Maclaurin series expansion of
y
y
y
.
Browse the problem library