hw6-solutions

# 2 4t problem 8 let us consider the dierential

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: on 3.5. Problem 7: Let us consider the diﬀerential equation y ′ (t) = 1 −4 y (t) 4 −7 with initial condition 3 . 2 y (0) = We ﬁrst compute the eigenvalues of A: 0 = det λ−1 4 = (λ − 1)(λ + 7) + 16 = λ2 + 6λ + 9. −4 λ + 7 Therefore, A has a single eigenvalue of multiplicity two (λ = −3). Applying the solution formula discussed in lecture gives y (t) = eλt (1 − λt) y (0) + t Ay (0) = e−3t (1 + 3t) = e−3t 3 −5 +t 2 −2 3 + 4t . 2 + 4t Problem 8: Let us consider the diﬀerential equation y ′ (t) = −5 2 3 −2 with initial condition y (0) = 3 2 1 2 y(t) 3 . −1 We ﬁrst compute the eigenvalues of A: 0 = det λ+ 3 2 5 2 3 5 −2 = λ+ λ− 1 2 2 λ− 1 9 + = λ2 + 2λ + 1. 2 4 Therefore, A has a single eigenvalue of multiplicity two (λ = −1). Applying the solution formula discussed in lecture gives y (t) = eλt (1...
View Full Document

## This note was uploaded on 02/09/2014 for the course MATH 53 taught by Professor Staff during the Winter '08 term at Stanford.

Ask a homework question - tutors are online