Explanation Definition used: The first order linear equation is of the form d y d t + p ( t ) y = g ( t ) where p and g are function of the independent variable t . Formula used: The integrating factor μ ( t ) of the first order differential equation d y d t + p ( t ) y = g ( t ) is given as μ ( t ) = exp ∫ p ( t ) d t and the solution of the differential equation is given by y = 1 μ ( t ) [ ∫ μ ( s ) g ( s ) d s + c ] . Calculation: The given differential equation is y ′ + 1 2 y = 2 cos t with the initial value y ( 0 ) = − 1 . By comparing the equation y ′ + 1 2 y = 2 cos t with its standard form d y d t + p ( t ) y = g ( t ) , it is obtained that p ( t ) = 1 2 and g ( t ) = 2 cos t . Compute the integrating factor μ ( t ) of the differential equation y ′ + 1 2 y = 2 cos t as follows. μ ( t ) = exp ∫ p ( t ) d t = exp ∫ 1 2 d t = e t 2 Now the solution of the differential equation y ′ + 1 2 y = 2 cos t can be obtained by substituting the integrating factor in the formula y = 1 μ ( t ) [ ∫ μ ( s ) g ( s ) d s + c ] as follows. y = 1 μ ( t ) [ ∫ μ ( s ) g ( s ) d s + c ] = 1 e t 2 [ ∫ e t 2 ( 2 cos t ) d t + c ] = e − t 2 [ 2 ∫ e t

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 2.1, Problem 27P

To determine

The coordinates of the first local maximum point of the solution of the initial value differential equation y′+12y=2cost, y(0)=−1.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.1, Problem 26P

Chapter 2.1, Problem 28P

Students have asked these similar questions

Task: Matrix Theory: Singular Value Decomposition (SVD) Refer to Question 19 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing

Task: 2 Abstract Algebra: Rings and Ideals Refer to Question 22 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing 23 Dynamical Systems: Stability of Fixed Points Task: Refer to Question 23 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing

30 Laplace Transform: Differential Equation Solutions Task: Refer to Question 30 in the provided document. Link: https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing

Chapter 2 Solutions

Elementary Differential Equations

Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - Prob. 3P Ch. 2.1 - Prob. 4P Ch. 2.1 - Prob. 5P Ch. 2.1 - Prob. 6P Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - Prob. 8P Ch. 2.1 - Prob. 9P Ch. 2.1 - Prob. 10P

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

The coordinates of the first local maximum point of the solution of the initial value differential equation y ′ + 1 2 y = 2 cos t , y ( 0 ) = − 1 .

Concept explainers

Videos

The coordinates of the first local maximum point of the solution of the initial value differential equation y′+12y=2cost, y(0)=−1.

Want to see the full answer?

Chapter 2 Solutions