Find a general solution to the given differential equation. y'' + 7y' - 18y=0 A general solution is y(t) = C₁ e 2t + C₂ e - 9t
Find a general solution to the given differential equation. y'' + 7y' - 18y=0 A general solution is y(t) = C₁ e 2t + C₂ e - 9t
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
![**Solving Differential Equations**
**Objective:**
Find a general solution to the given differential equation.
**Equation:**
\[ y'' + 7y' - 18y = 0 \]
**General Solution:**
A general solution to the given differential equation is:
\[ y(t) = c_1 e^{2t} + c_2 e^{-9t} \]
Where:
- \( c_1 \) and \( c_2 \) are arbitrary constants.
- \( e \) represents the base of the natural logarithm.
- \( t \) is the independent variable (often representing time).
**Description:**
This solution involves finding the characteristic equation of the differential equation and solving for its roots. The exponential terms \( e^{2t} \) and \( e^{-9t} \) are derived from these roots, and they form the basis of the general solution combined with arbitrary constants.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feff5a5e2-b243-453e-8506-ee83e62a3be5%2F27e18c70-f2e0-42a5-be6a-51c20cb95b7b%2F3d43a4e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving Differential Equations**
**Objective:**
Find a general solution to the given differential equation.
**Equation:**
\[ y'' + 7y' - 18y = 0 \]
**General Solution:**
A general solution to the given differential equation is:
\[ y(t) = c_1 e^{2t} + c_2 e^{-9t} \]
Where:
- \( c_1 \) and \( c_2 \) are arbitrary constants.
- \( e \) represents the base of the natural logarithm.
- \( t \) is the independent variable (often representing time).
**Description:**
This solution involves finding the characteristic equation of the differential equation and solving for its roots. The exponential terms \( e^{2t} \) and \( e^{-9t} \) are derived from these roots, and they form the basis of the general solution combined with arbitrary constants.
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