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I've also found a recursion relation for the equation to be:

An+2 = -An / (n+2)(n+1)

I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this?

Any help would be greatly appreciated!

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- Thread starter Sam D
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- #1

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I've also found a recursion relation for the equation to be:

An+2 = -An / (n+2)(n+1)

I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this?

Any help would be greatly appreciated!

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vela

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<Moderator's note: Approved as it is more than two weeks since the OP has been seen. Member has been warned not to post full solutions. This is an exception as it closes the thread.>

Let's start from your reccurence relation:

##A_{n+2}=\frac{-A_N}{(n+2)(n+1)}##

First collect even ##n## values:

##A_2= \frac{-A_0}{2.1}##

##A_4= \frac{-A_2}{4.3}=\frac{A_0}{4.3.2.1}=\frac{A_0}{4!}##

##A_6= \frac{-A_4}{6.5}=\frac{-A_0}{6.5.4.3.2.1}=\frac{-A_0}{6!}##

.........

Now take odd n values:

##A_3= \frac{-A_1}{3.2}##;

##A_5= \frac{-A_3}{5.4}=\frac{A_1}{5.4.3.2.1}=\frac{A_0}{5!}##

##A_7= \frac{-A_5}{7.6}=\frac{-A_0}{7.6.5.4.3.2.1}=\frac{-A_0}{7!}##

.........

So final solution is

On putting the values of ##A_n## in Maclaurin series solution (##y(x)=\sum_{n=0}{ A_n x^n}##),

##y(x)=\sum_{n=0}{ \frac{(-1)^n x^{2n}}{ 2n!} + \frac{(-1)^n x^{2n+1}}{ 2n+1!}}##

##y(x)= A_0 \cos (x) + A_1 \sin (x)##

Let's start from your reccurence relation:

##A_{n+2}=\frac{-A_N}{(n+2)(n+1)}##

First collect even ##n## values:

##A_2= \frac{-A_0}{2.1}##

##A_4= \frac{-A_2}{4.3}=\frac{A_0}{4.3.2.1}=\frac{A_0}{4!}##

##A_6= \frac{-A_4}{6.5}=\frac{-A_0}{6.5.4.3.2.1}=\frac{-A_0}{6!}##

.........

Now take odd n values:

##A_3= \frac{-A_1}{3.2}##;

##A_5= \frac{-A_3}{5.4}=\frac{A_1}{5.4.3.2.1}=\frac{A_0}{5!}##

##A_7= \frac{-A_5}{7.6}=\frac{-A_0}{7.6.5.4.3.2.1}=\frac{-A_0}{7!}##

.........

So final solution is

On putting the values of ##A_n## in Maclaurin series solution (##y(x)=\sum_{n=0}{ A_n x^n}##),

##y(x)=\sum_{n=0}{ \frac{(-1)^n x^{2n}}{ 2n!} + \frac{(-1)^n x^{2n+1}}{ 2n+1!}}##

##y(x)= A_0 \cos (x) + A_1 \sin (x)##

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