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 [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

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

Publisher: Addison Wesley

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

3/4+1/2=

if a=2 and b=1 1) Calculate 49(B-1)2+7B−1AT+7ATB−1+(AT)2 2)Find a matrix C such that (B − 2C)-1=A 3) Find a non-diagonal matrix E ̸= B such that det(AB) = det(AE)

1.2.17. (!) Let G,, be the graph whose vertices are the permutations of (1,..., n}, with two permutations a₁, ..., a,, and b₁, ..., b, adjacent if they differ by interchanging a pair of adjacent entries (G3 shown below). Prove that G,, is connected. 132 123 213 312 321 231

Chapter 2 Solutions

Fundamentals of Differential Equations [With CDROM] - 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.