- #1
latentcorpse
- 1,444
- 0
What is the general solution of
[itex]x^2y''+xy'-y=0[/itex]
i tried a series solution [itex]y=\sum_{n=0}^{\infty} a_n x^n[/itex]
and whitteld it down to
[itex]\sum_{n=0}^{\infty} (n^2-1) a_n x^n=0[/itex] but this isn't getting me anywhere
secondly how do i show that x=0 is a regular point of
[itex](1-x^2)y''-2xy'+2y=0[/itex]
the definition of a regular point is that :
x0 is a regular point of y''+a1(x)y'+a0(x)y=0 if the coefficients a1 and a0 can be expressed as convergent power series in (x-x0) for this value of x0.
so if i rearrange our equation into the desired form i get that
[itex]a_1(x)=\frac{-2x}{1-x^2},a_2(x)=\frac{2}{1-x^2}[/itex]
for x0=0, a1=0 and a2=2.
what do i say to round of my argument. something like:
"clearly 0 and 2 can be expressed as a convergent power series in x" or
"clearly a1 and a0 arent singular for this value of x0"
how do i round this off?
[itex]x^2y''+xy'-y=0[/itex]
i tried a series solution [itex]y=\sum_{n=0}^{\infty} a_n x^n[/itex]
and whitteld it down to
[itex]\sum_{n=0}^{\infty} (n^2-1) a_n x^n=0[/itex] but this isn't getting me anywhere
secondly how do i show that x=0 is a regular point of
[itex](1-x^2)y''-2xy'+2y=0[/itex]
the definition of a regular point is that :
x0 is a regular point of y''+a1(x)y'+a0(x)y=0 if the coefficients a1 and a0 can be expressed as convergent power series in (x-x0) for this value of x0.
so if i rearrange our equation into the desired form i get that
[itex]a_1(x)=\frac{-2x}{1-x^2},a_2(x)=\frac{2}{1-x^2}[/itex]
for x0=0, a1=0 and a2=2.
what do i say to round of my argument. something like:
"clearly 0 and 2 can be expressed as a convergent power series in x" or
"clearly a1 and a0 arent singular for this value of x0"
how do i round this off?