Solve the given initial-value problem. y(x) = y" + 4y + 4y = (7 + x)e-2x, y(0) = 5, y'(0) = 8
Solve the given initial-value problem. y(x) = y" + 4y + 4y = (7 + x)e-2x, y(0) = 5, y'(0) = 8
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
![**Problem Statement: Initial-Value Problem**
Solve the given initial-value problem.
\[ y'' + 4y' + 4y = (7 + x)e^{-2x}, \quad y(0) = 5, \quad y'(0) = 8 \]
\[ y(x) = \boxed{\quad} \]
**Explanation:**
The problem is a second-order linear non-homogeneous differential equation with given initial conditions. The solution involves finding a function \( y(x) \) that satisfies the differential equation and the initial conditions provided.
1. **Differential Equation**:
\[ y'' + 4y' + 4y = (7 + x)e^{-2x} \]
This is the main equation that needs to be solved.
2. **Initial Conditions**:
\[ y(0) = 5, \quad y'(0) = 8 \]
These conditions specify the values of the function \( y(x) \) and its first derivative \( y'(x) \) at \( x = 0 \).
**Steps to Solve**:
1. Solve the corresponding homogeneous equation.
2. Find a particular solution to the non-homogeneous equation.
3. Apply the initial conditions to determine the constants in the general solution.
The box below the equation indicates that the final answer for \( y(x) \) should be written there once the problem is solved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c333d55-482b-4cad-8212-73eff6844637%2F088a1c89-2baa-4b04-9dce-2d6ef1f408fc%2Fhf1tjv_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Initial-Value Problem**
Solve the given initial-value problem.
\[ y'' + 4y' + 4y = (7 + x)e^{-2x}, \quad y(0) = 5, \quad y'(0) = 8 \]
\[ y(x) = \boxed{\quad} \]
**Explanation:**
The problem is a second-order linear non-homogeneous differential equation with given initial conditions. The solution involves finding a function \( y(x) \) that satisfies the differential equation and the initial conditions provided.
1. **Differential Equation**:
\[ y'' + 4y' + 4y = (7 + x)e^{-2x} \]
This is the main equation that needs to be solved.
2. **Initial Conditions**:
\[ y(0) = 5, \quad y'(0) = 8 \]
These conditions specify the values of the function \( y(x) \) and its first derivative \( y'(x) \) at \( x = 0 \).
**Steps to Solve**:
1. Solve the corresponding homogeneous equation.
2. Find a particular solution to the non-homogeneous equation.
3. Apply the initial conditions to determine the constants in the general solution.
The box below the equation indicates that the final answer for \( y(x) \) should be written there once the problem is solved.
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