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4. Solve the following initial value problems using the Laplace transform: a) y + 3y = 0, y(0) = 1.5. b) y" - y - 6y=0, y(0)=11, y'(0) = 28 c) y" - 4y + 3y = 6l-8, y(0) = 0, y'(0) = 0 d) y" + 3y +2.25y = 91³ +64, y(0) = 1, y'(0) = 31.5 e) y" + 3y - 4y = 6e²-³, y(1.5) = 4, y′(1.5) = 5

4. Solve the following initial value problems using the Laplace transform: a) y + 3y = 0, y(0) = 1.5. b) y" - y - 6y=0, y(0)=11, y'(0) = 28 c) y" - 4y + 3y = 6l-8, y(0) = 0, y'(0) = 0 d) y" + 3y +2.25y = 91³ +64, y(0) = 1, y'(0) = 31.5 e) y" + 3y - 4y = 6e²-³, y(1.5) = 4, y′(1.5) = 5

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Only need help with part e. Thank you

**Problem 4: Solving Initial Value Problems Using the Laplace Transform** In this section, we explore initial value problems. Our task is to solve each differential equation using the Laplace transform method. Here are the given problems: a) \( y' + 3y = 0, \quad y(0) = 1.5 \). b) \( y'' - y' - 6y = 0, \quad y(0) = 11, \, y'(0) = 28 \). c) \( y'' - 4y' + 3y = 6t - 8, \quad y(0) = 0, \, y'(0) = 0 \). d) \( y'' + 3y' + 2.25y = 9t^3 + 64, \quad y(0) = 1, \, y'(0) = 31.5 \). e) \( y'' + 3y' - 4y = 6e^{2t - 3}, \quad y(1.5) = 4, \, y'(1.5) = 5 \). **Instructions:** For each problem, apply the Laplace transform to convert the differential equations into algebraic equations. Solve these algebraic equations for the Laplace transform of the solution, and then use the inverse Laplace transform to find the explicit solution to the initial value problem.
Transcribed Image Text:**Problem 4: Solving Initial Value Problems Using the Laplace Transform** In this section, we explore initial value problems. Our task is to solve each differential equation using the Laplace transform method. Here are the given problems: a) \( y' + 3y = 0, \quad y(0) = 1.5 \). b) \( y'' - y' - 6y = 0, \quad y(0) = 11, \, y'(0) = 28 \). c) \( y'' - 4y' + 3y = 6t - 8, \quad y(0) = 0, \, y'(0) = 0 \). d) \( y'' + 3y' + 2.25y = 9t^3 + 64, \quad y(0) = 1, \, y'(0) = 31.5 \). e) \( y'' + 3y' - 4y = 6e^{2t - 3}, \quad y(1.5) = 4, \, y'(1.5) = 5 \). **Instructions:** For each problem, apply the Laplace transform to convert the differential equations into algebraic equations. Solve these algebraic equations for the Laplace transform of the solution, and then use the inverse Laplace transform to find the explicit solution to the initial value problem.
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