The auxiliary equation for the given differential equation has complex roots. Find a general solution. y''-4y' +29y=0 cos 5t + C₂ e 2t sin 5t y(t)= C₁ e 2t
The auxiliary equation for the given differential equation has complex roots. Find a general solution. y''-4y' +29y=0 cos 5t + C₂ e 2t sin 5t y(t)= C₁ e 2t
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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![### Solving Differential Equations with Complex Roots
#### Problem Statement:
The auxiliary equation for the given differential equation has complex roots. Find a general solution.
Given differential equation:
\[ y'' - 4y' + 29y = 0 \]
#### General Solution:
The general solution for the differential equation with complex roots is given by:
\[ y(t) = c_1 e^{2t} \cos 5t + c_2 e^{2t} \sin 5t \]
In this equation:
- \( c_1 \) and \( c_2 \) are constants that can be determined based on initial conditions.
- \( e^{2t} \) represents the exponential growth factor.
- \( \cos 5t \) and \( \sin 5t \) represent the oscillatory components of the solution.
This solution is derived based on the characteristic polynomial of the differential equation which yields complex roots, leading to this particular form of the general solution. This form is commonly encountered in second-order linear differential equations with constant coefficients where the discriminant of the characteristic equation is negative.
### Explanation:
The provided solution highlights the standard method for solving second-order linear differential equations with complex roots. The components of the solution reflect the interplay between exponential growth and oscillatory behavior, a typical scenario in scenarios such as damped harmonic motion or certain electrical circuits.
For educational purposes, it is important to understand the steps leading to this solution:
1. Determine the characteristic equation from the given differential equation.
2. Solve for the roots of the characteristic equation.
3. Use the nature of the roots (complex in this case) to form the general solution.
This method allows for a systematic approach to finding solutions for differential equations encountered in various fields of science and engineering.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feff5a5e2-b243-453e-8506-ee83e62a3be5%2F64db8cfa-9cd2-4a97-9a35-47fd3e50ffff%2Fnnsdcif_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Differential Equations with Complex Roots
#### Problem Statement:
The auxiliary equation for the given differential equation has complex roots. Find a general solution.
Given differential equation:
\[ y'' - 4y' + 29y = 0 \]
#### General Solution:
The general solution for the differential equation with complex roots is given by:
\[ y(t) = c_1 e^{2t} \cos 5t + c_2 e^{2t} \sin 5t \]
In this equation:
- \( c_1 \) and \( c_2 \) are constants that can be determined based on initial conditions.
- \( e^{2t} \) represents the exponential growth factor.
- \( \cos 5t \) and \( \sin 5t \) represent the oscillatory components of the solution.
This solution is derived based on the characteristic polynomial of the differential equation which yields complex roots, leading to this particular form of the general solution. This form is commonly encountered in second-order linear differential equations with constant coefficients where the discriminant of the characteristic equation is negative.
### Explanation:
The provided solution highlights the standard method for solving second-order linear differential equations with complex roots. The components of the solution reflect the interplay between exponential growth and oscillatory behavior, a typical scenario in scenarios such as damped harmonic motion or certain electrical circuits.
For educational purposes, it is important to understand the steps leading to this solution:
1. Determine the characteristic equation from the given differential equation.
2. Solve for the roots of the characteristic equation.
3. Use the nature of the roots (complex in this case) to form the general solution.
This method allows for a systematic approach to finding solutions for differential equations encountered in various fields of science and engineering.
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