Explanation Given: The initial value problem is given as, d y d x + 2 y = Q ( x ) … ( 1 ) And the initial condition is y ( 0 ) = 0 . Approach: The general solution to the linear differential equation is given by the steps as follows: Step 1 Write the equation in Standard Form as, d y d x + P ( x ) y = Q ( x ) . Step 2 Calculate the integrating factor μ ( x ) by the formula, μ ( x ) = e ∫ P ( x ) d x . Step 3 the general solution is given as, y × μ ( x ) = ∫ Q ( x ) μ ( x ) d x + C . Here C is any arbitrary constant. Calculation: For 0 ≤ x ≤ 3 , the value is Q = 2 . Substitute 2 for Q in equation ( 1 ) as, d y d x + 2 y = 2 Integrating factor is given as, e ∫ P ( x ) d x = e ∫ 2 d x = e 2 x Therefore, the general solution is given as, y × e 2 x = ∫ 2 e 2 x d x + c y e 2 x = 2 ∫ e 2 x d x + c y e 2 x = 2 ( e 2 x 2 ) + c y e 2 x = e 2 x + c y = 1 + c e − 2 x Therefore, the general solution is, y = 1 + c e − 2 x … ( 2 ) The initial condition is y ( 0 ) = 0 . Substitute 0 for x and 0 for y in equation ( 2 ) as, y = 1 + c e − 2 x 0 = 1 + c e 0 0 = 1 + c c = − 1 Substitute the value of c in equation ( 2 ) as, y = 1 − e − 2 x Thus, solution is given as, y = 1 − e − 2 x … ( 3 ) For x > 3 , the value is Q = − 2 . Substitute − 2 for Q in equation ( 1 ) as, d y d x + 2 y = − 2 Integrating factor is given as, e ∫ P ( x ) d x = e ∫ 2 d x = e 2 x Therefore, the general solution is given as,

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

7th Edition

ISBN: 9780321977182

Author: Nagle, R. Kent, Saff, Edward B., Snider, Arthur David

Publisher: PEARSON

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 2.3, Problem 32E

To determine

To find:

The solution to the initial value problem for 0≤x≤3 and x>3.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.3, Problem 31E

Chapter 2.3, Problem 33E

Students have asked these similar questions

Given: Circle J 2 What is the value of y? A. 38 C. 68 B. 50 D. 92

please find the answers for the yellows boxes using the information and the picture below

Consider the function f(x) = x²-1. (a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative. Show all your steps clearly. (b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the graph where x 1 and x-> 1+h (for a small positive value of h, illustrate conceptually). Then, draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the value you found in part (a). (c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in the context of the graph of f(x). How does the rate of change of this function vary at different points?

Chapter 2 Solutions

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

Ch. 2.2 - Prob. 1E Ch. 2.2 - In Problems 1-6, determine whether the given...Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E

Knowledge Booster

Learn more about

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

The solution to the initial value problem for 0 ≤ x ≤ 3 and x > 3 .

Solution Summary: The author explains how the solution to the linear differential equation is given by the steps as follows: Write the equation in Standard Form and calculate the integrating factor.

Concept explainers

Videos

To find:

The solution to the initial value problem for 0≤x≤3 and x>3.

Want to see the full answer?

Chapter 2 Solutions

The solution to the initial value problem for 0 ≤ x ≤ 3 and x &gt; 3 .

Solution Summary: The author explains how the solution to the linear differential equation is given by the steps as follows: Write the equation in Standard Form and calculate the integrating factor.

Concept explainers

Videos

To find: The solution to the initial value problem for 0≤x≤3 and x>3.

Want to see the full answer?

Chapter 2 Solutions

The solution to the initial value problem for 0 ≤ x ≤ 3 and x > 3 .

To find:

The solution to the initial value problem for 0≤x≤3 and x>3.