Abstract:

This lecture (Vorlesung 13) by Prof. Karl-Heinz Lotze from the Friedrich-Schiller-Universität Jena introduces the "Variation of Constants" method as a second technique for finding a particular solution to inhomogeneous linear second-order ordinary differential equations with constant coefficients. The lecture systematically derives the formulas for the functions u'(x) and v'(x) used in the ansatz y_p(x) = u(x)y_1(x) + v(x)y_2(x), where y_1 and y_2 are fundamental solutions to the corresponding homogeneous equation. The method involves setting up a system of two algebraic equations for u' and v', solving them, and then integrating to find u and v. The lecture demonstrates the application of this method using the same example differential equation (y'' + y' - 2y = e^(2x)) that was solved in the previous lecture using a different technique, arriving at the same particular solution y_p = (1/4)e^(2x).

Summary of "Variation of Constants" for Solving Linear Second-Order ODEs

• 0:00 Introduction and Recap: The lecture begins by recalling that the general solution of an inhomogeneous linear second-order ODE is the sum of the general solution of the homogeneous equation (y_h = C_1y_1 + C_2y_2) and a particular solution (y_p) of the inhomogeneous equation. The goal is to find methods for determining y_p.
• 1:24 Method of Variation of Constants Introduced: This lecture focuses on the "Variation of Constants" method as the second approach to find y_p.
• 2:04 Steps of the Method:
  • 2:16 Step 1: Solve the Homogeneous Equation: Find the fundamental solutions y_1(x) and y_2(x) for the homogeneous equation ay'' + by' + cy = 0. The homogeneous solution is y_h(x) = C_1y_1(x) + C_2y_2(x).
  • 3:19 Step 2: Ansatz for Particular Solution: Vary the constants C_1 and C_2 by replacing them with functions u(x) and v(x). The ansatz for the particular solution is y_p(x) = u(x)y_1(x) + v(x)y_2(x).
• 4:41 Substitution and Derivation:
  • The ansatz y_p and its derivatives y_p' and y_p'' are substituted into the original inhomogeneous differential equation ay'' + by' + cy = f(x).
  • 11:31 Imposing an Auxiliary Condition (Eichung): To simplify the resulting complex equation and to create a system for u' and v', an auxiliary condition is imposed: u'(x)y_1(x) + v'(x)y_2(x) = 0. This choice simplifies the expression for y_p'.
  • 14:14 Resulting System of Equations: This condition, along with the simplified main equation, leads to a system of two linear algebraic equations for u'(x) and v'(x):
    1. u'(x)y_1(x) + v'(x)y_2(x) = 0
    2. u'(x)y_1'(x) + v'(x)y_2'(x) = f(x)/a
• 16:38 Solving for u'(x) and v'(x):
  • The system is solved for u'(x) and v'(x), yielding:
    • u'(x) = - (1/a) * f(x) * y_2(x) / W(y_1, y_2)
    • v'(x) = (1/a) * f(x) * y_1(x) / W(y_1, y_2)
  • 18:42 Wronskian Determinant: The denominator W(y_1, y_2) = y_1y_2' - y_2y_1' is the Wronskian determinant of the fundamental solutions.
• 18:48 Finding u(x) and v(x): u(x) and v(x) are found by integrating u'(x) and v'(x) respectively. Integration constants are typically omitted as we only need a particular solution.
• 19:32 Example Application: The method is applied to the differential equation y'' + y' - 2y = e^(2x).
  • 20:31 Homogeneous Solution: y_1 = e^x, y_2 = e^(-2x). (Here a=1, b=1, c=-2)
  • 21:30 Derivatives: y_1' = e^x, y_2' = -2e^(-2x).
  • 21:51 Wronskian: W(y_1, y_2) = -3e^(-x).
  • 22:32 Inhomogeneity: f(x) = e^(2x).
  • 22:52 Calculating u'(x): u'(x) = - (1/1) * e^(2x) * e^(-2x) / (-3e^(-x)) = (1/3)e^x.
  • 23:48 Calculating v'(x): v'(x) = (1/1) * e^(2x) * e^x / (-3e^(-x)) = -(1/3)e^(4x).
  • 24:42 Integrating to find u(x) and v(x):
    • u(x) = (1/3)e^x.
    • v(x) = -(1/12)e^(4x).
  • 25:51 Constructing Particular Solution (y_p):
    • y_p = u(x)y_1(x) + v(x)y_2(x) = (1/3)e^x * e^x + (-1/12)e^(4x) * e^(-2x)
    • y_p = (1/3)e^(2x) - (1/12)e^(2x) = (1/4)e^(2x).
• 27:07 Final General Solution: The general solution is y(x) = C_1e^x + C_2e^(-2x) + (1/4)e^(2x), which matches the result obtained by the previous method.
• 28:01 Outlook: The next lecture will introduce another method for finding particular solutions, which can be faster for specific forms of the inhomogeneity f(x).