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Differential equations power series method

  • Thread starter SpiffyEh
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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= [tex]\sum a_{n}t^{n}[/tex] from n=0 to infinity
y'= [tex]\sum na_{n}t^{n-1}[/tex] 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[tex]_{1}[/tex]+2a[tex]_{2}[/tex]t+3a[tex]_{3}[/tex]t[tex]^{2}[/tex]+[tex]\sum[na_{n}[/tex]+a[tex]_{n-4}][/tex]t[tex]^{n-1}[/tex] = 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[tex]_{n}[/tex] = -a[tex]_{n-4}[/tex]/n whre n >= 4

But I don't understand what i do with the other part.. the a[tex]_{1}[/tex]+2a[tex]_{2}[/tex]t+3a[tex]_{3}[/tex]t[tex]^{2}[/tex]
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
 

Answers and Replies

  • #2
vela
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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= [tex]\sum a_{n}t^{n}[/tex] from n=0 to infinity
y'= [tex]\sum na_{n}t^{n-1}[/tex] 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[tex]_{1}[/tex]+2a[tex]_{2}[/tex]t+3a[tex]_{3}[/tex]t[tex]^{2}[/tex]+[tex]\sum[na_{n}[/tex]+a[tex]_{n-4}][/tex]t[tex]^{n-1}[/tex] = 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[tex]_{n}[/tex] = -a[tex]_{n-4}[/tex]/n whre n >= 4

But I don't understand what i do with the other part.. the a[tex]_{1}[/tex]+2a[tex]_{2}[/tex]t+3a[tex]_{3}[/tex]t[tex]^{2}[/tex]
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 [itex]a_1=a_2=a_3=0[/itex]. 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.
 

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