Why do we use different methods for finding power series solutions?

[matematikuvol]

Why sometimes we search solution of power series in the way:
$$y(x)=\sum^{\infty}_{n=0}a_nx^n$$
and sometimes
$$y(x)=\sum^{\infty}_{n=0}a_nx^{n+1}$$?

 

[tiny-tim]

hi matematikuvol!

Why sometimes we search solution of power series in the way:
$$y(x)=\sum^{\infty}_{n=0}a_nx^n$$
and sometimes
$$y(x)=\sum^{\infty}_{n=0}a_nx^{n+1}$$?

no particular reason …

sometimes one gives neater equations than the other …

they'll both work (provided, of course, that y(0) = 0)

 

[matematikuvol]

I think that in the case when
$$\alpha(x)y''(x)+\beta(x)y'(x)+\gamma(x)y(x)=0$$
if $\alpha(0)=0$ you must work with $\sum^{\infty}_{n=0}a_nx^{n+k}$, but I'm not sure.

 

[tiny-tim]

I think that in the case when
$$\alpha(x)y''(x)+\beta(x)y'(x)+\gamma(x)y(x)=0$$
if $\alpha(0)=0$ you must work with $\sum^{\infty}_{n=0}a_nx^{n+k}$, but I'm not sure.

but that's the same as $\sum^{\infty}_{n=0}b_nx^n$ with $b_n = a_{n-k}$ for n ≥ k, and $b_n = 0$ otherwise

 

[blue_raver22]

Power series solutions are a common approach used in mathematics and physics to solve differential equations. They are particularly useful when the differential equation cannot be solved using traditional methods such as separation of variables or substitution. The use of different methods for finding power series solutions is based on the specific characteristics of the differential equation being solved.

One reason for using different methods is to ensure that the power series solution is valid for the given differential equation. For example, some differential equations have singular points, where the solution may not be defined or may become infinite. In these cases, it is necessary to use a power series solution that is valid at these singular points.

Another reason for using different methods is to simplify the solution process. In some cases, using a power series in the form of y(x)=\sum^{\infty}_{n=0}a_nx^n may lead to a simpler solution than using y(x)=\sum^{\infty}_{n=0}a_nx^{n+1}. This is because the latter form involves an extra variable and may result in more complicated calculations.

Additionally, the choice of power series form may depend on the initial conditions of the differential equation. For example, if the initial condition is given at x=0, it may be more convenient to use y(x)=\sum^{\infty}_{n=0}a_nx^n as it will result in a simpler expression for the coefficients.

Furthermore, certain differential equations may have different types of solutions depending on the power series form used. For instance, the solution obtained using y(x)=\sum^{\infty}_{n=0}a_nx^{n+1} may have different convergence properties compared to the solution obtained using y(x)=\sum^{\infty}_{n=0}a_nx^n. Therefore, the choice of power series form can affect the overall behavior of the solution.

In summary, the use of different methods and power series forms for finding solutions is necessary to ensure the validity and simplicity of the solution, as well as to account for different initial conditions and convergence properties. As a scientist, it is important to carefully consider these factors when choosing the appropriate method for solving a given differential equation.