x=0 is a regular singular point of the given DE. Show that the indicial roots of the singularity do not differ by an integer. Use the the method of Frobenius to obtain two linearly independent series solutions about x=0. Form the general solution on (0,\infty)

Question

anonymous · Answer

$$2xy''-y'+2y=0$$

anonymous · Answer

frobenius theorem states

$$(x-x_0)^r\sum_{n=0}^\infty c_n(x-x_0)^n=\sum_{n=0}^\infty c_n(x-x_0)^{n+r}$$

anonymous · Answer

@experimentX

anonymous · Answer

Question when they take the derivatives the place at which the sums start are all n=0 however when you did it before they went up one as you took the derivative

anonymous · Answer

is all n=0 since it's arbitrary until we know what r =

anonymous · Answer

@zepp

anonymous · Answer

@zzr0ck3r

anonymous · Answer

in other words if you had

$$y=\sum_{n=0}^\infty c_nx^{n-1}$$

if you plug it in you get $$x^{-1}$$ which would not be  a power series correct?

anonymous · Answer

so they move it up one whereas here we have

n+r-1 but we do not know what r is

anonymous · Answer

@eliassaab

anonymous · Answer

$$y=\sum_{n=0}^\infty c_nx^{n+r}$$

$$y'=\sum_{n=0}^\infty c_n(n+r)x^{n+r-1}$$

anonymous · Answer

$$y''=\sum_{n=0}^\infty c_n(n+r-1)(n+r)x^{n+r-2}$$

anonymous · Answer

$$2x\sum_{n=0}^\infty c_n (n+r-1)(n+r)x^{n+r-2}-\sum_{n=0}^\infty c_n(n+r)x^{n+r-1} $$

$$+2\sum_{n=0}^\infty c_nx^{n+r}$$

anonymous · Answer

$$\sum_{n=0}^\infty2c_n(n+r-1)(n+r)x^{n+r-1}-\sum_{n=0}^\infty c_n(n+r)x^{n+r-1}$$

$$+\sum_{n=0}^\infty 2c_nx^{n+r}$$

anonymous · Answer

$$\sum_{n=0}^\infty c_n(2n+2r-2)(n+r)x^{n+r-1}-\sum_{n=0}^\infty c_n(n+1)x^{n+r-1}$$

anonymous · Answer

$$\sum_{n=0}^\infty [(2n+2r-2)(n+r)-n+r]c_nx^{n+r-1}$$

anonymous · Answer

for last sum let

$$k=n+1$$

$$k-1=n$$

and first added sums n=k
$$\sum_{k=0}^\infty [2k+2r-2)(k+r)-k+r]c_k x^{k+r-1}+\sum_{k=1}^\infty2c_{k-1}x^{k+r-1}$$

experimentx · Answer

is not so fond of power series solution. I can provide answer from Maple if you want.

anonymous · Answer

pull out nonzero termof first sum

$$[(2r-2)(r)+r]c_0x^{r-1}+\sum_{n=1}^\infty(2k+2r-2)(k+r)-k+r]c_kx^{k+r-1}$$

$$+\sum_{k=1}^\infty 2c_{k-1}x^{k+r-1}$$

anonymous · Answer

gah i need to resub -.-

anonymous · Answer

these problems are driving me a-wall