Changing indices in ODE series solution
$begingroup$
I am given the following 2nd order ODE (in Sturm-Liouville form):
$$(u'e^{-x^2/2})'=-lambda e^{-x^2/2}$$
I am asked to try polynomial solutions $U_lambda(x)$ of degree $lambda$. I therefore substitute the general form of a polynomial of degree $lambda$,
$$U_lambda(x)=sum_{n=0}^lambda c_nx^n,$$ along with its derivatives $$U'_lambda(x)=sum_{n=1}^lambda nc_nx^{n-1}quadtextrm{and}quad U''_lambda(x)=sum_{n=2}^lambda n(n-1)c_nx^{n-2}$$ into the ODE. This gives
$$sum_{n=2}^lambda n(n-1)c_nx^{n-2}-sum_{n=1}^lambda nc_nx^{n}+sum_{n=0}^lambda lambda c_nx^n=0.$$
I can trivially extend the lower limit of the second sum to $n=0$ thanks to the $n$ term. My issue lies in the first summation. I was taught to change index such that $n-2to n$, as to align the powers of $x$.
According to the solution sheet I was given, this yields
$$sum_{n=0}^lambda (n+2)(n+1)c_{n+2}x^{n}-sum_{n=0}^lambda nc_nx^{n}+sum_{n=0}^lambda lambda c_nx^n=0$$
which gives the desired recurrence relation
$$c_{n+2}=frac{n-lambda}{(n+2)(n+1)}c_n.$$
My issue lies in the index change ($nto n+2$) in the first summation. Shouldn't the upper limit of the sum also change, from $n=lambda$ to $n=lambda-2$? The lower limit changes (from $n=2$ to $n=0$), so surely the upper limit would have to change too? And if it did change, wouldn't that make it illegal to group the coefficients of $x^n$ and therefore to derive the recurrence relation?
I have encountered similar index changes e.g. in Frobenius' method, but there the upper limit was usually $infty$ so I wasn't too bothered by having to it leave it unchanged. But since $lambda$ here is finite, I am not sure if I am missing something.
Thanks.
ordinary-differential-equations power-series
asked Dec 19 '18 at 16:26
martinmartin
136
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add a comment |
$begingroup$
I am given the following 2nd order ODE (in Sturm-Liouville form):
$$(u'e^{-x^2/2})'=-lambda e^{-x^2/2}$$
I am asked to try polynomial solutions $U_lambda(x)$ of degree $lambda$. I therefore substitute the general form of a polynomial of degree $lambda$,
$$U_lambda(x)=sum_{n=0}^lambda c_nx^n,$$ along with its derivatives $$U'_lambda(x)=sum_{n=1}^lambda nc_nx^{n-1}quadtextrm{and}quad U''_lambda(x)=sum_{n=2}^lambda n(n-1)c_nx^{n-2}$$ into the ODE. This gives
$$sum_{n=2}^lambda n(n-1)c_nx^{n-2}-sum_{n=1}^lambda nc_nx^{n}+sum_{n=0}^lambda lambda c_nx^n=0.$$
I can trivially extend the lower limit of the second sum to $n=0$ thanks to the $n$ term. My issue lies in the first summation. I was taught to change index such that $n-2to n$, as to align the powers of $x$.
According to the solution sheet I was given, this yields
$$sum_{n=0}^lambda (n+2)(n+1)c_{n+2}x^{n}-sum_{n=0}^lambda nc_nx^{n}+sum_{n=0}^lambda lambda c_nx^n=0$$
which gives the desired recurrence relation
$$c_{n+2}=frac{n-lambda}{(n+2)(n+1)}c_n.$$
My issue lies in the index change ($nto n+2$) in the first summation. Shouldn't the upper limit of the sum also change, from $n=lambda$ to $n=lambda-2$? The lower limit changes (from $n=2$ to $n=0$), so surely the upper limit would have to change too? And if it did change, wouldn't that make it illegal to group the coefficients of $x^n$ and therefore to derive the recurrence relation?
I have encountered similar index changes e.g. in Frobenius' method, but there the upper limit was usually $infty$ so I wasn't too bothered by having to it leave it unchanged. But since $lambda$ here is finite, I am not sure if I am missing something.
Thanks.
ordinary-differential-equations power-series
asked Dec 19 '18 at 16:26
martinmartin
136
$endgroup$
add a comment |
$begingroup$
I am given the following 2nd order ODE (in Sturm-Liouville form):
$$(u'e^{-x^2/2})'=-lambda e^{-x^2/2}$$
I am asked to try polynomial solutions $U_lambda(x)$ of degree $lambda$. I therefore substitute the general form of a polynomial of degree $lambda$,
$$U_lambda(x)=sum_{n=0}^lambda c_nx^n,$$ along with its derivatives $$U'_lambda(x)=sum_{n=1}^lambda nc_nx^{n-1}quadtextrm{and}quad U''_lambda(x)=sum_{n=2}^lambda n(n-1)c_nx^{n-2}$$ into the ODE. This gives
$$sum_{n=2}^lambda n(n-1)c_nx^{n-2}-sum_{n=1}^lambda nc_nx^{n}+sum_{n=0}^lambda lambda c_nx^n=0.$$
I can trivially extend the lower limit of the second sum to $n=0$ thanks to the $n$ term. My issue lies in the first summation. I was taught to change index such that $n-2to n$, as to align the powers of $x$.
According to the solution sheet I was given, this yields
$$sum_{n=0}^lambda (n+2)(n+1)c_{n+2}x^{n}-sum_{n=0}^lambda nc_nx^{n}+sum_{n=0}^lambda lambda c_nx^n=0$$
which gives the desired recurrence relation
$$c_{n+2}=frac{n-lambda}{(n+2)(n+1)}c_n.$$
My issue lies in the index change ($nto n+2$) in the first summation. Shouldn't the upper limit of the sum also change, from $n=lambda$ to $n=lambda-2$? The lower limit changes (from $n=2$ to $n=0$), so surely the upper limit would have to change too? And if it did change, wouldn't that make it illegal to group the coefficients of $x^n$ and therefore to derive the recurrence relation?
I have encountered similar index changes e.g. in Frobenius' method, but there the upper limit was usually $infty$ so I wasn't too bothered by having to it leave it unchanged. But since $lambda$ here is finite, I am not sure if I am missing something.
Thanks.
ordinary-differential-equations power-series
asked Dec 19 '18 at 16:26
martinmartin
136
$endgroup$
I am given the following 2nd order ODE (in Sturm-Liouville form):
$$(u'e^{-x^2/2})'=-lambda e^{-x^2/2}$$
I am asked to try polynomial solutions $U_lambda(x)$ of degree $lambda$. I therefore substitute the general form of a polynomial of degree $lambda$,
$$U_lambda(x)=sum_{n=0}^lambda c_nx^n,$$ along with its derivatives $$U'_lambda(x)=sum_{n=1}^lambda nc_nx^{n-1}quadtextrm{and}quad U''_lambda(x)=sum_{n=2}^lambda n(n-1)c_nx^{n-2}$$ into the ODE. This gives
$$sum_{n=2}^lambda n(n-1)c_nx^{n-2}-sum_{n=1}^lambda nc_nx^{n}+sum_{n=0}^lambda lambda c_nx^n=0.$$
I can trivially extend the lower limit of the second sum to $n=0$ thanks to the $n$ term. My issue lies in the first summation. I was taught to change index such that $n-2to n$, as to align the powers of $x$.
According to the solution sheet I was given, this yields
$$sum_{n=0}^lambda (n+2)(n+1)c_{n+2}x^{n}-sum_{n=0}^lambda nc_nx^{n}+sum_{n=0}^lambda lambda c_nx^n=0$$
which gives the desired recurrence relation
$$c_{n+2}=frac{n-lambda}{(n+2)(n+1)}c_n.$$
My issue lies in the index change ($nto n+2$) in the first summation. Shouldn't the upper limit of the sum also change, from $n=lambda$ to $n=lambda-2$? The lower limit changes (from $n=2$ to $n=0$), so surely the upper limit would have to change too? And if it did change, wouldn't that make it illegal to group the coefficients of $x^n$ and therefore to derive the recurrence relation?
I have encountered similar index changes e.g. in Frobenius' method, but there the upper limit was usually $infty$ so I wasn't too bothered by having to it leave it unchanged. But since $lambda$ here is finite, I am not sure if I am missing something.
Thanks.
ordinary-differential-equations power-series
ordinary-differential-equations power-series
asked Dec 19 '18 at 16:26
martinmartin
136
asked Dec 19 '18 at 16:26
martinmartin
136
edited Dec 19 '18 at 17:05
martin
asked Dec 19 '18 at 16:26
martinmartin
136
asked Dec 19 '18 at 16:26
martinmartin
136
asked Dec 19 '18 at 16:26
martinmartin
136
136
add a comment |
add a comment |
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