Question

Y"-3xy=0 given ordinary point x=0. power series

Answer 1

`\displaystyle y=\sum_(n\ge0)a_nx^n`

`\implies\displaystyle y''=\sum_(n\ge2)n(n-1)a_nx^(n-2)`


`y''-3xy=0`

`\displaystyle\sum_(n\ge2)n(n-1)a_nx^(n-2)-3\sum_(n\ge0)a_nx^(n+1)=0`

`\displaystyle2a_2+\sum_(n\ge3)n(n-1)a_nx^(n-2)-3\sum_(n\ge0)a_nx^(n+1)=0`

`\displaystyle2a_2+\sum_(n\ge3)n(n-1)a_nx^(n-2)-3\sum_(n\ge3)a_(n-3)x^(n-2)=0`

`\displaystyle2a_2+\sum_(n\ge3)\bigg(n(n-1)a_n-3a_(n-3)\bigg)x^(n-2)=0`

This generates the recurrence relation


`\begin{cases}a_0=a_0\\a_1=a_1\\2a_2=0\\n(n-1)a_n-3a_(n-3)=0&\text{for }n\ge3\end{cases}`

Because you have


`n(n-1)a_n-3a_(n-3)=0\implies a_n=\frac3{n(n-1)}a_(n-3)`

it follows that
`a_2=0\implies a_5=a_8=a_(11)=\cdots=a_(n=3k-1)=0` for all
`k\ge1`.

For
`n=1,4,7,10,\ldots`, you have


`a_1=a_1`

`a_4=\frac3{4*3}a_1=(3*2)/(4!)a_1`

`a_7=\frac3{7*6}a_4=(3*5)/(7*6*5)=(3^2*5*2)/(7!)`

`a_(10)=\frac3{10*9}a_7=(3*8)/(10*9*8)a_7=(3^3*8*5*2)/(10!)a_1`

so that, in general, for
`n=3k-2`,
`k\ge1`, you have


`a_(n=3k-2)=(3^(k-1)\displaystyle\prod_(\ell=1)^(k-1)(3\ell-1))/((3k-2)!)a_1`

Now, for
`n=0,3,6,9,\ldots`, you have


`a_0=a_0`

`a_3=\frac3{3*2}a_0=\frac3{3!}a_0`

`a_6=\frac3{6*5}a_3=(3*4)/(6*5*4)a_3=(3^2*4)/(6!)a_0`

`a_9=\frac3{9*8}a_6=(3*7)/(9*8*7)a_6=(3^3*7*4)/(9!)a_0`

and so on, with a general pattern for
`n=3k`,
`k\ge0`, of


`a_(n=3k)=(3^k\displaystyle\prod_(\ell=1)^k(3\ell-2))/((3k)!)a_0`

Putting everything together, we arrive at the solution


`y=\displaystyle\sum_(n\ge0)a_nx^n`

`y=a_0\underbrace{\displaystyle\sum_(k\ge0)(3^k\displaystyle\prod_(\ell=1)^k(3\ell-2))/((3k)!)x^(3k)}_(n=0,3,6,9,\ldots)+a_1\underbrace{\displaystyle\sum_(k\ge1)(3^(k-1)\displaystyle\prod_(\ell=1)^(k-1)(3\ell-1))/((3k-2)!)x^(3k-2)}_(n=1,4,7,10,\ldots)`

To show this solution is sufficient, I've attached is a plot of the solution taking
`y(0)=a_0=1` and
`y'(0)=a_1=0`, with
`n=6`. (I was hoping to be able to attach an animation that shows the series solution (orange) converging rapidly to the exact solution (blue), but no such luck.)