DiffEQ 2nd order series sol'n problem: (1 - x)y'' + y = 0, x0 = 0

[VinnyCee]

This problem is from section 5.2 in Boyce, DiPrima's Differential Equations 8th edition.

$$(1 - x)\,y''\,+\,y\,=\,0$$

I get:

$$2\,a_2\,+\,a_0\,+\,\sum_{n\,=\,1}^{\infty}\,\left[(n\,+\,2)\,(n\,+\,1)\,a_{n\,+\,2}\,-\,n\,(n\,+\,1)\,a_{n\,+\,1}\,+\,a_n\right]\,x_n\,=\,0$$

Which leads to one equation:

$$2\,a_2\,+\,a_0\,=\,0$$

$$a_2\,=\,-\frac{1}{2}\,a_0$$

and the recursion formula:

$$a_{n\,+\,2}\,=\,\frac{n\,(n\,+\,1)\,a_{n\,+\,1}\,-\,a_n}{(n\,+\,2)\,(n\,+\,1)}$$

Now I am totally lost, just like in this other thread, because I don't know how to move on from this step. Please help

 

[VinnyCee]

What now?

Using the recursion formula above, i get the following:

$$a_{(0)+2} =\frac{((0)+1)(0)a_{(0)+1}-a_0}{((0)+2)((0)+1)}\:\:\:\:\:\:\:<br /> a_2 =\frac{-a_0}{2*1}\\$$

$$a_{(1)+2} =\frac{((1)+1)(1)a_{(1)+1}-a_{(1)}}{((1)+2)((1)+1)}\:\:\:\:\:\:\:<br /> a_3 =\frac{-a_0-a_1}{3*2}\\$$

$$a_{(2)+2} =\frac{((2)+1)(2)a_{(2)+1}-a_{(2)}}{((2)+2)((2)+1)}\:\:\:\:\:\:\:<br /> a_4 =\frac{-a_1-\frac{1}{2}a_0}{4*3}\\$$

But I don't see much of a pattern. What am I doing wrong? Please help, thank you in advance.

 

[VinnyCee]

Almost done...

After fixing the eroor in the first term of the recursion series above, I now set a0 = 1 and a1 = 0 to separate the two-types of terms left in the formula:

$$y(x)\,=\,\sum_{n\,=\,0}^{\infty}\,a_n\,x^n\,=\,a_0\,+\,a_1\,x\,-\,\frac{a_0}{2\,*\,1}\,x^2\,-\,\frac{a_0\,+\,a_1}{3\,*\,2}\,x^3\,-\,\frac{a_1\,+\,\frac{1}{2}\,a_0}{4\,*\,3}\,x^4$$

$$y_{a_0\,=\,1,\,a_1\,=\,0}(x)\,=\,(1)\,+\,(0)\,x\,-\,\frac{(1)}{2\,*\,1}\,x^2\,-\,\frac{(1)\,+\,(0)}{3\,*\,2}\,x^3\,-\,\frac{(0)\,+\,\frac{1}{2}\,(1)}{4\,*\,3}\,x^4\,=\,1\,-\,\frac{1}{2}\,x^2\,-\,\frac{1}{6}\,x^3\,-\,\frac{1}{24}\,x^4\,+\,...$$

Now with a0 = 0 and a1 = 1:

$$y_{a_0\,=\,0,\,a_1\,=\,1}(x)\,=\,(0)\,+\,(1)\,x\,-\,\frac{(0)}{2\,*\,1}\,x^2\,-\,\frac{(0)\,+\,(1)}{3\,*\,2}\,x^3\,-\,\frac{(1)\,+\,\frac{1}{2}\,(0)}{4\,*\,3}\,x^4\,=\,x\,-\,\frac{1}{6}\,x^3\,-\,\frac{1}{12}\,x^4\,-\,\frac{1}{24}\,x^5\,+\,...$$

Now, factoring out the a0 and a1 terms from each:

$$y(x)\,=\,a_0\,\left[1\,-\,\frac{1}{2}\,x^2\,-\,\frac{1}{6}\,x^3\,-\,\frac{1}{24}\,x^4\,+\,...\right]\,+\,a_1\,\left[x\,-\,\frac{1}{6}\,x^3\,-\,\frac{1}{12}\,x^4\,-\,\frac{1}{24}\,x^5\,+\,...\right]$$

The book says this is the answer, but is there any way to generalize this equation more? Possibly by using a summation?

 

[dextercioby]

U could use factorials,but what would be the purpose?You found the solution,i say that's enough.

Daniel.