Find the general solution of the given second-order differential equation. 20y" - 11y' – 3y = 0 y(x) =
Find the general solution of the given second-order differential equation. 20y" - 11y' – 3y = 0 y(x) =
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**
Find the general solution of the given second-order differential equation.
\[ 20y'' - 11y' - 3y = 0 \]
**Solution**
The general solution \( y(x) = \) [Blank for student input]
**Explanation**
This problem involves solving a linear homogeneous second-order differential equation with constant coefficients. The equation is given as:
\[ 20y'' - 11y' - 3y = 0 \]
To find the general solution, follow these steps:
1. **Characteristic Equation**: Assume a solution of the form \( y = e^{rx} \) and substitute into the differential equation to obtain the characteristic equation. Solve for \( r \) to find the roots.
2. **Roots of the Characteristic Equation**: The nature of the roots (real and distinct, real and repeated, or complex conjugates) will determine the form of the general solution.
3. **General Solution**: Based on the roots, write the general solution for \( y(x) \).
This equation is a standard topic in differential equations courses, exploring techniques in solving linear homogeneous equations with constant coefficients.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd39804f-6d08-4135-954a-c92ac4c2d043%2F1bd37463-2a5e-4b77-91ac-6abfef43831e%2Fav84ie_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem**
Find the general solution of the given second-order differential equation.
\[ 20y'' - 11y' - 3y = 0 \]
**Solution**
The general solution \( y(x) = \) [Blank for student input]
**Explanation**
This problem involves solving a linear homogeneous second-order differential equation with constant coefficients. The equation is given as:
\[ 20y'' - 11y' - 3y = 0 \]
To find the general solution, follow these steps:
1. **Characteristic Equation**: Assume a solution of the form \( y = e^{rx} \) and substitute into the differential equation to obtain the characteristic equation. Solve for \( r \) to find the roots.
2. **Roots of the Characteristic Equation**: The nature of the roots (real and distinct, real and repeated, or complex conjugates) will determine the form of the general solution.
3. **General Solution**: Based on the roots, write the general solution for \( y(x) \).
This equation is a standard topic in differential equations courses, exploring techniques in solving linear homogeneous equations with constant coefficients.
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