3. (a) (b) (c) Solve the homogeneous DEs. Show all work. y(4) +4y"-21y=0 y"+y+ 7y=0 2y"-3y'-2y=0 with initial conditions y(0)= 4, y'(0) = 0
3. (a) (b) (c) Solve the homogeneous DEs. Show all work. y(4) +4y"-21y=0 y"+y+ 7y=0 2y"-3y'-2y=0 with initial conditions y(0)= 4, y'(0) = 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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Question
![### Problem 3: Solve the Homogeneous Differential Equations (DEs)
#### Instructions:
Show all work for solving each differential equation.
---
#### (a) Fourth-Order Homogeneous DE
\[
y^{(4)} + 4y'' - 21y = 0
\]
---
#### (b) Second-Order Homogeneous DE
\[
y'' + y' + 7y = 0
\]
---
#### (c) Second-Order Homogeneous DE with Initial Conditions
\[
2y'' - 3y' - 2y = 0 \quad \text{with initial conditions} \quad y(0) = 4, \; y'(0) = 0
\]
---
### Explanation:
Each problem requires solving a differential equation. These are typical standard linear homogeneous differential equations with constant coefficients.
- Problem (a) is a fourth-order differential equation.
- Problems (b) and (c) are second-order differential equations, with problem (c) providing specific initial conditions to solve for particular solutions.
When solving these, you will typically look for characteristic equations, solve for the roots, and use these roots to construct general solutions. For problem (c), use the initial conditions to find specific solution constants.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc033cde6-9355-4f65-911a-aa4e14330b10%2F40793c07-0546-4442-940d-40cd3a41e857%2Fnclmjwa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 3: Solve the Homogeneous Differential Equations (DEs)
#### Instructions:
Show all work for solving each differential equation.
---
#### (a) Fourth-Order Homogeneous DE
\[
y^{(4)} + 4y'' - 21y = 0
\]
---
#### (b) Second-Order Homogeneous DE
\[
y'' + y' + 7y = 0
\]
---
#### (c) Second-Order Homogeneous DE with Initial Conditions
\[
2y'' - 3y' - 2y = 0 \quad \text{with initial conditions} \quad y(0) = 4, \; y'(0) = 0
\]
---
### Explanation:
Each problem requires solving a differential equation. These are typical standard linear homogeneous differential equations with constant coefficients.
- Problem (a) is a fourth-order differential equation.
- Problems (b) and (c) are second-order differential equations, with problem (c) providing specific initial conditions to solve for particular solutions.
When solving these, you will typically look for characteristic equations, solve for the roots, and use these roots to construct general solutions. For problem (c), use the initial conditions to find specific solution constants.
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