The point x = 0 is a regular singular point of the given differential equation. 6xy"\" y'+ 6y =0 Show that the indicial roots r of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on (0, ). Step 1 We are asked to find two series solutions to the following differential equation about the regular singular point x = 0. 6xy"\" y'+ 6y =0 If it was a case where the given point was an ordinary point, we would substitute y = 5. c.x'7 and solve for the coefficients. However, in n n=0 oO this case, x = 0 is a regular singular point, so we use Frobenius' Theorem with the substitution y = S cx? + rand solve for the n n=0 coefficients and r to find the solutions. oO Let y = + cyxn th n=0 We substitute this series and its derivatives into the given differential equation. We will then solve for two sets of c,, to find the two solutions. Note that Frobenius' Theorem does not always guarantee that there are two linearly independent solutions, but we will see shortly that two distinct relevant values of r will lead to distinct solutions. The given differential equation includes non-zero terms for the first and second derivative of y. We therefore need these derivatives of oO y= a cpyx? +f n=0 These derivatives always have the same form, so we refer to the equations in Example 2 of Section 6.3. co s (n+ rey? tro 4 n=0 y' y"= 5 (n+ Maer le? to 2 n=0 Substitute the power series into the given differential equation. oO By" y+ 6y = S (ne nin tr 1} ree | (( ntr + a ent) 7 n=0 n=0