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y" + 2y' + 2y = 5x² − 5 cos(2x); y(0) = 12, y'(0) = −2

y" + 2y' + 2y = 5x² − 5 cos(2x); y(0) = 12, y'(0) = −2

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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**Problem Statement: Solving an Initial Value Problem (IVP)** Solve the IVP: \[ y'' + 2y' + 2y = 5x^2 - 5\cos(2x) \] Subject to the initial conditions: \[ y(0) = 12, \quad y'(0) = -2 \] **Explanation:** This problem involves solving a second-order linear differential equation with constant coefficients. The equation is non-homogeneous due to the presence of \(5x^2 - 5\cos(2x)\) on the right-hand side. **Initial Conditions:** - \( y(0) = 12 \): The value of the function \( y \) at \( x = 0 \). - \( y'(0) = -2 \): The value of the first derivative of \( y \) at \( x = 0 \). These conditions are crucial for finding the specific solution to the differential equation that satisfies these constraints.
Transcribed Image Text:**Problem Statement: Solving an Initial Value Problem (IVP)** Solve the IVP: \[ y'' + 2y' + 2y = 5x^2 - 5\cos(2x) \] Subject to the initial conditions: \[ y(0) = 12, \quad y'(0) = -2 \] **Explanation:** This problem involves solving a second-order linear differential equation with constant coefficients. The equation is non-homogeneous due to the presence of \(5x^2 - 5\cos(2x)\) on the right-hand side. **Initial Conditions:** - \( y(0) = 12 \): The value of the function \( y \) at \( x = 0 \). - \( y'(0) = -2 \): The value of the first derivative of \( y \) at \( x = 0 \). These conditions are crucial for finding the specific solution to the differential equation that satisfies these constraints.
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