Assume the hypotheses of Exercise 3, and assume that y₁(x) and y₂(x) are both solutions to the firstorder linear equation satisfying the initial condition y(x₀) = y₀.

a. Verify that y(x) = y₁(x) - y₂(x) satisfies the initial value problem

b. For the integrating factor ν(x) = e^{∫P(x) dx}, show that

Conclude that ν(x)[y₁(x) - y₂(x)] K constant.

c. From part (a), we have y₁(x₀) - y₂(x₀) = 0. Since ν(x) > 0 for a 1(x) - y₂(x) = 0 on the interval (a, b). Thus y₁(x) = y₂(x) for all a

Exercise 3

a. Assume that P(x) and Q(x) are continuous over the interval [a, b]. Use the Fundamental Theorem of Calculus, Part 1, to show that any function y satisfying the equation

for ν(x) = e^{∫P(x) dx} is a solution to the first-order linear equation.

b. If  then show that any solution y in part (a) satisfies the initial condition y(x₀) = y₀.

Transcribed Image Text:

y' + P(x)y = 0, y(x) = 0.