ODE Lecture Notes, Section 5.5

ODE Lecture Notes Section 5.5 Page 1 of 6

Section 5.5: Series Solutions Near a Regular Singular Point, Part I Big Idea: According to Frobenius, it is valid to assume a series solution of the form

r n y=( x - x0) an ( x - x 0 ) to a second-order linear differential equation near a regular singular n=0 point.

Big Skill: You should be able to apply the method of Frobenius to solve second-order linear differential equation near regular singular points.

Recall: If x = x0 is a regular singular point of the second order linear equation P( x) yⅱ+ Q( x) y + R( x) y = 0 , Q( x) R( x) then limx= lim xp( x) = finite and limx2 = lim xq( x) = finite .   x x0P( x) x x 0 x x0P( x) x x 0

n 2 n This means xp( x) = pn ( x - x0 ) and x q( x) = qn ( x - x0 ) are convergent for some radius n=0 n=0 about x0. 2 2 Thus, we can write the original equation as: x yⅱ+臌轾 xp( x) xy +臌轾 x q( x) y = 0

If all the coefficients pn and qn are zero except for p0 and q0, then the equation reduces to 2 x yⅱ+ p0 xy + q 0 y = 0 , which is an Euler equation.

Since the equation we are trying to solve looks like an equation with “Euler coefficients” times power series, we will look for a solution that is of the form of an “Euler solution” times a power series:

ゥ r n r+ n y=( x - x0) 邋 an( x - x 0) = a n ( x - x 0 ) n=0 n = 0

As part of the solution, we must determine:  The values of r that make this a valid solution.

 The recurrence relation for the coefficients an.  The radius of convergence.

The theory behind a solution of this form is due to Frobenius. ODE Lecture Notes Section 5.5 Page 2 of 6

To solve a linear second order equation near a regular singular point using the method of Frobenius:  Identify singular points and verify they are regular.

r+ n  Assume a solution of y= an ( x - x0 ) and its derivatives: n=0

r+ n -1 r+ n -2 y= an ( r + n)( x - x0 ) , yⅱ= an ( r + n)( r + n -1)( x - x0 ) n=0 n=0  Substitute the assumed solution and its derivatives into the given equation.

o You may have to re-write coefficients in terms of ( x- x0 ) … r+ n  Shift indices so that all series solutions have ( x- x0 ) in the general term.  “Spend” any terms needed so that all series start at the same index value. o This should result in a term that looks like the characteristic equation of the corresponding Euler equation. o This is called the indicial equation. o The roots of the indicial equation are called the exponents of the singularity. r+ n  Set the coefficient of ( x- x0 ) to zero to get the recurrence relation.  Use the recurrence relation with each exponent to get the general term for each of the two solutions.  Compute the radius of convergence for each solution. ODE Lecture Notes Section 5.5 Page 3 of 6

Practice: 1. Solve the differential equation 2x2 yⅱ- xy +( 1 + x) y = 0 . ODE Lecture Notes Section 5.5 Page 4 of 6 ODE Lecture Notes Section 5.5 Page 5 of 6

2. Find the solution of the Legendre equation near the point x = -1: (1-x2 ) yⅱ - 2 xy +a( a + 1) y = 0 . ODE Lecture Notes Section 5.5 Page 6 of 6