MATH 312 Section 6.2: Series Solutions about Singular Points Prof. Jonathan Duncan Walla Walla University Spring Quarter, 2008
Outline 1 Classifying Singular Points 2 The Method of Frobenius 3 Conclusions
Regular and Irregular Singular Points Recall that a point x 0 is called ordinary if P(x) and Q(x) from the equation y + P(x)y + Q(x)y = 0 are both analytic at x 0. A point which is not ordinary is called singular. Definition 6.2 A singular point x 0 is said to be a regular singular point of the differential equation y + P(x)y + Q(x)y = 0 if the functions (x x 0 )P(x) and (x x 0 ) 2 Q(x) are both analytic at x 0. A singular point which is not regular is said to be an irregular singular point of the equation. Note: Practically speaking, if (x x 0 ) appears at most once as a factor in the denominator of P(x) and at most twice as a factor in the denominator of Q(x), then x = x 0 is a regular singular point.
An Example Let s examine a differential equation and classify singular points. Example Determine the singular points of the differential equation shown below and classify them as regular or irregular. (x 2 9) 2 y + (x + 3)y + 2y = 0 y + P(x) = (x + 3) (x + 3) 2 (x 3) 2 y 2 + (x + 3) 2 (x 3) 2 y = 0 1 (x + 3)(x 3) 2 Q(x) = 2 (x + 3) 2 (x 3) 2 x = 3 is irregular x = 3 is regular
Frobenius Theorem One reason that we distinguish between regular and irregular singular points is the following theorem. Theorem 6.2 If x = x 0 is a regular singular point of the differential equation y + P(x)y + Q(x)y = 0, then there exists at least one solution of the form: y = (x x 0 ) r c n (x x 0 ) n = c n (x x 0 ) n+r where the number r is a constant to be determined. The series will converge at least on some interval 0 < x x 0 < R. Note: Unlike ordinary points, singular points do not necessarily have a pair of independent solutions. We are only guaranteed one solution.
The Method of Frobenius To put this theorem into practice, we need a straight-forward method to both find r and solve the differential equation. Method of Frobenius To find a series solution about a regular singular point x 0, use the following procedure. 1 Substitute y(x) = c n(x x 0 ) n+r into the equation. 2 Use the identity property to find an equation for r, called the indicial equation. 3 Solve the indicial equation to find the values of r called the indicial roots or exponent of the singular point. 4 Use these roots to generate the c n four the power series.
An Example We now put this method into practice to solve a differential equation about a singular point. Example Find a solution to 2xy y + 2y = 0 about the regular singular point x = 0. X y = c nx n+r X y = c n(n + r)x n+r 1 X y = c n(n + r)(n + r 1)x n+r 2 x r! X c 0 (2r 2 3r)x 1 + [2c n+1 (n + r + 1)(n + r) c n+1 (n + r + 1) + 2c n] x n = 0 r = 0 : c 1 = 2c 0, c 2 = 2c 0, c 3 = 4 9 c 0 r = 3 2 : c 1 = 2 5 c 0, c 2 = 2 35 c 0, c 3 = 4 945 c 0 y = c 1 1 + 2x + wx 2 + 4 «9 x3 + + c 2 x 3 2 1 2 5 x + 2 35 x2 4 «945 x3 +
Indicial Equations When we work with second order differential equations, the indicial equation will be quadratic, and will have three types of solutions. Indicial Equation Solutions The indicial equation for a second order differential equation will have roots r 1 r 2 which can be divided into three cases. Case I: r 1 and r 2 are distinct real roots where r 1 r 2 not a natural number. Case II: r 1 and r 2 are distinct real roots where r 1 r 2 is a natural number. Case III: r 1 and r 2 are equal. Note: As r is real exponent, we will not find complex solutions to the indicial equation.
Exploring Case I We now consider each of these three cases separately, starting with the first. Case I: r 1 r 2 Not a Natural Number If the indicial roots r 1 > r 2 are such that r 1 r 2 is not a natural number, then we have two distinct solutions: y 1 (x) = c n x n+r 1 y 2 (x) = n=1 c n x n+r 2 n=1 Note: In this case we are guaranteed two linearly independent series solutions of the form given above.
Exploring Case II We now examine the second possible situation. Case II: r 1 r 2 is a Natural Number If the indicial roots r 1 > r 2 are such that r 1 r 2 is a natural number, then there are two distinct solutions of the form: y 1 (x) = c n x n+r 1 y 2 (x) = Cy 1 (x) ln x + b n x n+r 2 Note: In case II, only if C = 0 will we have two series solutions. We will not now this until we have carefully examined the recurrence relations for the two indicial roots.
Wrapping it Up Finally, we consider the last case for our indicial roots. Case III: r 1 = r 2 If the two indicial roots are equal, we have two linearly independent solutions of the form: y 1 = c n x n+r 1 y 2 (x) = y 1 (x) ln x + b n x n+r 1 Note: The second solution for Case III and Case II can be found using the reduction of order term. n=1
Important Concepts Things to Remember from Section 6.2 1 Classification of Singular points as: regular singular points irregular singular points 2 Using the method of Frobenius to find series solutions about regular singular points. 3 The three possible cases when solving a quadratic indicial equation and the forms of the solutions they produce.