# Differential equations power series method

## Homework Statement

using the power series method (centered at t=0) y'+t^3y = 0 find the recurrence relation

## Homework Equations

y= $$\sum a_{n}t^{n}$$ from n=0 to infinity
y'= $$\sum na_{n}t^{n-1}$$ from n=1 to infinity

## The Attempt at a Solution

I went through and solved by putting the values from b into the equation and i got down to this:
a$$_{1}$$+2a$$_{2}$$t+3a$$_{3}$$t$$^{2}$$+$$\sum[na_{n}$$+a$$_{n-4}]$$t$$^{n-1}$$ = 0

the 1 2 and 3 should be subscripted on a

I understand how to get an which i think i got right... i got a$$_{n}$$ = -a$$_{n-4}$$/n whre n >= 4

But I don't understand what i do with the other part.. the a$$_{1}$$+2a$$_{2}$$t+3a$$_{3}$$t$$^{2}$$
Do i set it to 0 and solve for something? or do i set each individual component to 0. If i do ti this way and i find the power series solution by going through values of n until i find a pattern they all end up being 0 which can't be right. I'm completely lost.

Please let me know, this is due tomorrow and I have everything done except I don't understand what to do with this one. I just need to know what to do with that part, I can do the rest.

Thank you so much

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## Homework Statement

using the power series method (centered at t=0) y'+t^3y = 0 find the recurrence relation

## Homework Equations

y= $$\sum a_{n}t^{n}$$ from n=0 to infinity
y'= $$\sum na_{n}t^{n-1}$$ from n=1 to infinity

## The Attempt at a Solution

I went through and solved by putting the values from b into the equation and i got down to this:
a$$_{1}$$+2a$$_{2}$$t+3a$$_{3}$$t$$^{2}$$+$$\sum[na_{n}$$+a$$_{n-4}]$$t$$^{n-1}$$ = 0

the 1 2 and 3 should be subscripted on a

I understand how to get an which i think i got right... i got a$$_{n}$$ = -a$$_{n-4}$$/n whre n >= 4

But I don't understand what i do with the other part.. the a$$_{1}$$+2a$$_{2}$$t+3a$$_{3}$$t$$^{2}$$
Do i set it to 0 and solve for something? or do i set each individual component to 0. If i do ti this way and i find the power series solution by going through values of n until i find a pattern they all end up being 0 which can't be right. I'm completely lost.

Please let me know, this is due tomorrow and I have everything done except I don't understand what to do with this one. I just need to know what to do with that part, I can do the rest.

Thank you so much
You're right. You set the first three terms to zero, so you get $a_1=a_2=a_3=0$. The only non-zero terms in your power series will therefore be for n=0, 4, 8, ....

This differential equation is separable, so you can find the solution in closed form. Expand it as a series and see if it matches what the series solution seems to be giving you.