Asked by philip
determine a series if solution of the equation y"(x)+x.y'(x)+y(x)=0
show me step plz
show me step plz
Answers
Answered by
Steve
review the conditions for the series to exist. In this case, it will exist for x=0, so we want
y = ∑a<sub>n</sub>x^n
y' = ∑na<sub>n</sub>x^(n-1)
y" = ∑n(n-1)a<sub>n</sub>x^(n-2)
The DE then says
∑n(n-1)a<sub>n</sub>x^(n-2) + x*∑na<sub>n</sub>x^(n-1) + ∑a<sub>n</sub>x^n = 0
∑n(n-1)a<sub>n</sub>x^(n-2) + ∑na<sub>n</sub>x^n + ∑a<sub>n</sub>x^n = 0
∑n(n-1)a<sub>n</sub>x^(n-2) + ∑(n+1)a<sub>n</sub>x^n = 0
Now shift y" so the powers of x add up
∑(n+2)(n+1)a<sub>n+2</sub>x^n + ∑(n+1)a<sub>n</sub>x^n = 0
∑[(n+2)(n+1)a<sub>n+2</sub> + (n+1)a<sub>n</sub>]x^n = 0
For the series to be zero for all x, we have the recurrence relation
(n+2)(n+1)a<sub>n+2</sub> + (n+1)a<sub>n</sub> = 0
a<sub>n+2</sub> = -a<sub>n</sub>/(n+2)
At this point, just start listing the coefficients and see where you go. There is a similar DE used as an example at
http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx
y = ∑a<sub>n</sub>x^n
y' = ∑na<sub>n</sub>x^(n-1)
y" = ∑n(n-1)a<sub>n</sub>x^(n-2)
The DE then says
∑n(n-1)a<sub>n</sub>x^(n-2) + x*∑na<sub>n</sub>x^(n-1) + ∑a<sub>n</sub>x^n = 0
∑n(n-1)a<sub>n</sub>x^(n-2) + ∑na<sub>n</sub>x^n + ∑a<sub>n</sub>x^n = 0
∑n(n-1)a<sub>n</sub>x^(n-2) + ∑(n+1)a<sub>n</sub>x^n = 0
Now shift y" so the powers of x add up
∑(n+2)(n+1)a<sub>n+2</sub>x^n + ∑(n+1)a<sub>n</sub>x^n = 0
∑[(n+2)(n+1)a<sub>n+2</sub> + (n+1)a<sub>n</sub>]x^n = 0
For the series to be zero for all x, we have the recurrence relation
(n+2)(n+1)a<sub>n+2</sub> + (n+1)a<sub>n</sub> = 0
a<sub>n+2</sub> = -a<sub>n</sub>/(n+2)
At this point, just start listing the coefficients and see where you go. There is a similar DE used as an example at
http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx
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