6) Solve the following problem, provide a well explained, correct and readable solution. ### Differential Equation ProblemGiven the initial value problem (IVP):\[ x'' + 6x' + 5x = 30 e^{-5t}, \]with initial conditions\[ x(0) = 8, \quad x'(0) = 28. \]Solve the IVP and find $ x(0.25) $ to at least four decimal places of accuracy. Enter your answer in the answer box.---### ExplanationThis problem involves solving a second-order linear differential equation with constant coefficients. The equation has a non-homogeneous part, $ 30 e^{-5t} $, which requires determining the particular solution. Initial conditions are provided to find the specific solution. The task demands accuracy to four decimal places for the value at $ t = 0.25 $.

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Description: 6) Solve the following problem, provide a well explained, correct and readable solution. ### Differential Equation ProblemGiven the initial value problem (IVP):\[ x'' + 6x' + 5x = 30 e^{-5t}, \]with initial conditions\[ x(0) = 8, \quad x'(0) = 28. \]

Consider the given IVP, x''+6x'+5x=30e-5t, x0=8, x'0=28. The homogeneous differential equation is…

Answered: -5t x" +6x' +5x = 30 e ", x(0) = 8,…

Math

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-5t x" +6x' +5x = 30 e ", x(0) = 8, x'(0) = 28 Solve the IVP and find x(0.25) to at least four decimal places of accuracy. Put x(0.25) in the answer box.

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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Question

6) Solve the following problem, provide a well explained, correct and readable solution.

### Differential Equation Problem

Given the initial value problem (IVP):

\[ x'' + 6x' + 5x = 30 e^{-5t}, \]

with initial conditions

\[ x(0) = 8, \quad x'(0) = 28. \]

Solve the IVP and find $ x(0.25) $ to at least four decimal places of accuracy. Enter your answer in the answer box.

---

### Explanation

This problem involves solving a second-order linear differential equation with constant coefficients. The equation has a non-homogeneous part, $ 30 e^{-5t} $, which requires determining the particular solution. Initial conditions are provided to find the specific solution. The task demands accuracy to four decimal places for the value at $ t = 0.25 $.

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