Definition: A p oint t hat i s n ot a n o rdinary p oint o f t he d ifferential equation a2 (x)y ′′ + a1 (x)y ′ + a0 (x)y = 0 is said to be a singular point of the differential equation. Definition: A s ingular p oint x = x 0 i s s aid t o b e a r egular s ingular p oint o f t he differential equation a2 (x)y ′′ + a1 (x)y ′ + a0 (x)y = 0 if the functions p(x) = (x − x0 )P (x) and a0 (x) q(x) = (x − x0 )2 Q(x) are both analytic at x0 (where P (x) = aa12 (x) (x) and Q(x) = a2 (x) . Definition: A singular point that is not regular is said to be an irregular singular point of the equation. Fact: If x − x0 appears at most to the first power in the denominator of P (x) and at most to the second power in the denominator of Q(x), then x = x0 is a regular singular point. Exercise 1. Classify the singular points of the equation (x2 − 1)2 y ′′ (x) + (x + 1)y ′ (x) − y(x) = 0. (Nagle Section 8.6 Example 1) Exercise 2. Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. (#2,4) 1 (a) x(x + 3)2 y ′′ − y = 0 (b) y ′′ − x1 y ′ + (x−1) 3y = 0 Class Exercise 1. Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. (#6,8,10) (a) x2 (x − 5)2 y ′′ + 4xy ′ + (x2 − 25)y = 0 (b) x(x2 + 1)2 y ′′ + y = 0 3 2 2 ′′ 2 ′ (c) (x − 2x − 3x) y + x(x − 3) y − (x + 1)y = 0 Frobenius’ Theorem: If x = x0 is a regular singular point of the differential equation a2 (x)y ′′ + a1 (x)y ′ + a0 (x)y = 0, then there exists at least one solution of the form P∞ P∞ y = (x − x0 )r n=0 cn (x − x0 )n = n=0 cn (x − x0 )n+r , where the number r is a constant to be determined. The series will converge at least on some interval 0 < x − x0 < R. Definition: The method of Frobenius generates an indicial equation. The roots of the indicial equation are called the indicial roots. What the Roots Tell Us About the Solutions Suppose the indicial roots, r1 and r2 , for the differential equation are real. Case 1: If r1 and r2 are distinct and the difference r1 − r2 is not a positive integer, then there exist two linearly independent solutions of the equation a2 (x)y ′′ + a1 (x)y ′ + a0 (x)y = 0 of the form P∞ P∞ y1 (x) = n=0 cn xn+r1 , c0 ̸= 0, and y2 (x) = n=0 bn xn+r2 , b0 ̸= 0. Case 2: If r1 and r2 are distinct and the difference r1 − r2 is a positive integer, then there exist two linearly independent solutions of the equation a2 (x)y ′′ + a1 (x)y ′ + a0 (x)y = 0 of the form P∞ P∞ y1 (x) = n=0 cn xn+r1 , c0 ̸= 0, and y2 (x) = Cy1 (x) ln x + n=0 bn xn+r2 , b0 ̸= 0, where C is a constant that could be zero. Case 3: If r1 and r2 are equal, then there exist two linearly independent solutions of the equation a2 (x)y ′′ + a1 (x)y ′ + a0 (x)y = 0 of the form P∞ P∞ y1 (x) = n=0 cn xn+r1 , c0 ̸= 0, and y2 (x) = y1 (x) ln x + n=1 bn xn+r1 . Exercise 3. Find a series solution about the regular singular point x = 0 of x2 y ′′ (x) − xy ′ (x) + (1 − x)y(x) = 0, x > 0. (Nagle Section 8.6 Example 4) Exercise 4. Find a series solution about the regular singular point x = 0 of xy ′′ (x) + 4y ′ (x) − xy(x) = 0, x > 0. (Nagle Section 8.6 Example 5) Exercise 5. Find a series solution about the regular singular point of x = 0 of xy ′′ (x) + 3y ′ (x) − xy(x) = 0, x > 0. (Nagle Section 8.6 Example 6) Exercise 6. x = 0 is a regular singular point of the given differential equation. Show that the indicial roots 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, ∞). (#16, 18) (a) 2xy ′′ + 5y ′ + xy = 0 (b) 2x2 y ′′ − xy ′ + (x2 + 1)y = 0 Class Exercise 2. x = 0 is a regular singular point of the given differential equation. Show that the indicial roots 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, ∞). (#20, 22, 24, 26) (a) x2 y ′′ − (x − 92 )y = 0 (b) x2 y ′′ + xy ′ + (x2 − 49 )y = 0 2 ′′ ′ (c) 2x y + 3xy + (2x − 1)y = 0 (d) x2 y ′′ + xy ′ + (x2 − 14 )y = 0 Exercise 7. x = 0 is a regular singular point of the differential equation y ′′ + x3 y ′ − 2y = 0. Show that the indicated roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0. Use the equation R − R P (x)dx y2 (x) = y1 (x) e y2 (x) dx 1 when necessary and a CAS, if instructed, to find a second solution. Form the general solution on (0, ∞). (#28) Class Exercise 3. x = 0 is a regular singular point of the differential equation xy ′′ + y ′ + y = 0. Show that the indicated roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0. Use the equation R − R P (x)dx y2 (x) = y1 (x) e y2 (x) dx 1 when necessary and a CAS, if instructed, to find a second solution. Form the general solution on (0, ∞). (#30) Homework: 1, 5, 9, 11, 13-29 (every 4th)