Question

Consider the second-order initial value problem, y′′ − 3y ′ + 2y = 0, y(0) = 0,y ′(0) = 1,0 ≤ x ≤ 1

Make a suitable substitution and show that the above equation can be written as a system of first order initial value problems.

Answer 1

To transform a second-order initial value problem into a first-order system, introduce a new variable u to represent y′. This results in two first-order equations, with the original initial conditions applying to these new variables.

Step-by-step explanation:

To convert the given second-order initial value problem into a set of first-order initial value problems, we introduce new variables to represent the first derivative of y. Let's set u = y′, which implies that u′ = y′′.

Substituting these into the given equation y′′ - 3y′ + 2y = 0, we get u′ - 3u + 2y = 0. Now, we can rewrite this as two first-order equations:

1. u′ = 3u - 2y
2. y′ = u

With initial conditions given by y(0) = 0 and u = y′(0) = 1, we have a set of first-order initial value problems defined over the interval 0 ≤ x ≤ 1.