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,

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ISBN: 9780135902738

Author: Nagle

Publisher: PEARSON

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Chapter 2.3, Problem 32E

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

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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.

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

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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.

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

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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.