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# differential equations

Mathematics

Homework

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( )
( )
( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
2
2 2 1
2 1
2
2 2 1
2
0 2 1
2
0 2 1
1 0
1 1 0
1 1 0
2 1 1 0
2 1 1 0
n n
n n
n n
n n n
n n n
n n n
n n n
n n n
n n n
n
n n n
n n n
x y xy
x n n a x x na x
n n a x n n a x na x
n n a x n n a x na x
n n a n n a na x
= =
= = =
+
= = =
+
= = =
=
=
=
+ + =
+ + =
At this point we could strip out some terms to get all the series starting at 2,
but that’s actually more work than is needed.
Let’s instead note that we could start the third series at 0
if we wa
n
n
=
=
nted to because that term is just zero.
Likewise the terms in the first series are zero for both 1 and 0 and
so we could start that series at 0.
If we do this all the series will now start at
n n
n
= =
=
0
and we can add them up without stripping terms out of any series.
n =
( ) ( ) ( )
( ) ( )
( ) ( )
2
2 2
2
2 2
2
0
0
0
0
0
2 1 1
3 2
3 02
n n n
n n n
n
n
n
n
n
n
n
n
n n a n n a na
n n a n n a na
n n a n n n a
x
x
x
=
+
+
=
=
+
+ + =
+
=
+
+ +
=
+
( ) ( )
( ) ( )
( )
2 2
2
2 2
2
2
2
0
2
0
Now set coefficients equal t
3
o zero
2
3 2 0, n 0,1, 2,...
3 2
n n
n n
n
n
n
n
n n a n a
n n a n a
n a
a
x
n n
+
+
+
=
+ +
+ + = =
=
+
=
+

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2
1
3
2 2
4
Plug in values of
0, 0
1,
6
4
2, 0
12 3
n
n a
a
n a
a a
n a
= =
= =
= = = =
( )
( )
1
3
1
5
2 4 5
0 1 2 3 5
4 5
1 1
1
4
5
1
9
9 3
6
3,
20 20 40
...
3
0 0 ...
6 40
3
6 40
a
a a
n a
y x a a x a x a x a x
a a
a x x x
x
y x a x x
÷
= = = =
= + + + + +
= + + + + +
= + +
÷
( ) ( )
( )
( )
1
3
4
1
0 0, 0 1
0 3
0 0 0
6 40
0 0
4 15
1
6 40
y y
y a
x
y x a x
= =
= + +
÷
=
= + +
÷
( )
( )
( )
3
4
1
1
1
4
5
2 3
1
3 8
0
1
3
Therefore,
6 40
x
y x a x
y a
a
x
y x x x
= + +
÷
=
=
= + +
÷

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Anonymous
I was struggling with this subject, and this helped me a ton!

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