Power Series & Singular Points: Why Change the Form?

[CPL.Luke]

when finding a power series solution we have to put the differential equation

ay''+by'+c=0

into the form

y''+By+C=0

this leads to singular points when a=0 but why can't we leave the equation in its original form and use power series substitution to avoid singular points?

or in other words why do we care about the second form?

 

[Hurkyl]

Experiment. Try it out and see. Take a simple example,

x y'' - 1 = 0

for example, and see what happens.

 

[tacman]

The reason we change the form of the differential equation when finding a power series solution is because it allows us to identify and analyze singular points more easily. Singular points are points where the coefficients of the differential equation become infinite, which can cause issues in the convergence of the power series solution. By transforming the equation into the form y''+By+C=0, we can clearly see when a=0 and therefore predict where singular points may occur.

Moreover, the second form of the equation also allows us to apply certain convergence tests, such as the ratio test, to determine the radius of convergence of the power series solution. This is important because it tells us up to what values of x our solution will be valid.

Additionally, transforming the equation into the second form also simplifies the calculations involved in finding the coefficients of the power series solution. This is because the coefficients are now only dependent on the values of B and C, rather than a, b, and c, which can be more complex and time-consuming to work with.

In summary, changing the form of the differential equation to y''+By+C=0 when finding a power series solution allows us to better understand and predict the behavior of the solution, as well as simplifying the calculations involved. It is a useful technique that helps us avoid potential issues with singular points and ensure the convergence of our solution.