1. Section 4.7 Variation of Parameters

2. METHOD OF VARIATION OF PARAMETERS For a second-order linear equation in standard form y″ + Py′ + Qy = g(x). 1.Find the complementary function y c (x) = c 1 y 1 (x) + c 2 y 2 (x). 2.Replace the constants by the functions u 1 and u 2 to form the particular solution y p (x) = u 1 (x)y 1 (x) + u 2 (x)y 2 (x). 3.Solve the following system of equations for u′ 1 and u′ 2 4.Integrate to find u 1 (x) and u 2 (x).

3. USING CRAMER’S RULE TO SOLVE THE SYSTEM The system of equations can be solved using Cramer’s Rule (determinants).

4. COMMENTS ON VARIATION OF PARAMETERS 1.The system of equations can be solved by other methods as well. That is, by substitution or elimination. 2.The method of Variation of Parameters is not limited to g(x) being either a polynomial, exponential, sine, cosine, or finite sums and products of these functions.

5. HIGHER-ORDER EQUATIONS Variation of Parameters can be used to solve higher-order equations. Let y p = u 1 (x)y 1 (x) + u 2 (x)y 2 (x) +... + u n (x)y n (x) Solve the following system of equations.

6. HOMEWORK 1–33 odd