Please solve the following, provide well explained and readable solution. ### Transcription for Educational Website---**Differential Equation Problem**Solve the initial value problem (IVP) given by the differential equation:\[ x'' + 6x' + 5x = 10t + 30e^{-2t} \]with the initial conditions: \[ x(0) = 2, \quad x'(0) = 8 \]**Task:**Find the value of $ x(0.25) $ to at least four decimal places of accuracy. Enter your answer in the provided answer box.---### Explanation of Elements:**Equation Explanation:**- $ x'' $: The second derivative of $ x $ with respect to $ t $, indicating the acceleration.- $ 6x' $: The first derivative of $ x $ (velocity) multiplied by 6.- $ 5x $: The function $ x $ itself multiplied by 5.- $ 10t + 30e^{-2t} $: The non-homogeneous part of the differential equation, representing a function dependent on $ t $ composed of a linear term and an exponential decay term.**Initial Conditions:**- $ x(0) = 2 $: The initial value of the function $ x $ at $ t = 0 $.- $ x'(0) = 8 $: The initial velocity of the function $ x $ at $ t = 0 $.**Objective:**Calculate $ x(0.25) $ accurately and input the result.---

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Description: Please solve the following, provide well explained and readable solution. ### Transcription for Educational Website---**Differential Equation Problem**Solve the initial value problem (IVP) given by the differential equation:\[ x'' + 6x' + 5x = 10t +

Concept Used:- If roots of characteristics equation is a, b where a≠b, real then general solution is…

Answered: -2t x" +6x' +5x = 10t + 30e 2", x(0) =…

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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Please solve the following, provide well explained and readable solution.

### Transcription for Educational Website

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**Differential Equation Problem**

Solve the initial value problem (IVP) given by the differential equation:

\[ x'' + 6x' + 5x = 10t + 30e^{-2t} \]

with the initial conditions:

\[ x(0) = 2, \quad x'(0) = 8 \]

**Task:**

Find the value of $ x(0.25) $ to at least four decimal places of accuracy. Enter your answer in the provided answer box.

---

### Explanation of Elements:

**Equation Explanation:**

- $ x'' $: The second derivative of $ x $ with respect to $ t $, indicating the acceleration.
- $ 6x' $: The first derivative of $ x $ (velocity) multiplied by 6.
- $ 5x $: The function $ x $ itself multiplied by 5.
- $ 10t + 30e^{-2t} $: The non-homogeneous part of the differential equation, representing a function dependent on $ t $ composed of a linear term and an exponential decay term.

**Initial Conditions:**

- $ x(0) = 2 $: The initial value of the function $ x $ at $ t = 0 $.
- $ x'(0) = 8 $: The initial velocity of the function $ x $ at $ t = 0 $.

**Objective:**

Calculate $ x(0.25) $ accurately and input the result.

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Expert Solution

Step 1

Concept Used:-

If roots of characteristics equation is a, b where a $\neq$ b, real

then general solution is $c_{1} e^{a t} + c_{2} e^{b t}$

Given:- x''+6x'+5x = 10t+30e^-2t ...(*)

x(0)=2, x'(0)=8

We first solve the homogeneous equation x''+6x'+5x = 0 ...(1)

So, characteristics equation for (1) is

$r^{2} + 6 r + 5 = 0 \Rightarrow (r + 1) (r + 5) = 0 \Rightarrow r = - 1, - 5$

Thus general solution is

$x (t) = c_{1} e^{- t} + c_{2} e^{- 5 t}$

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