2y' =√x + 2y + 1|
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
Solve the differential equation using v substitution:
![In this educational segment, we will discuss the differential equation displayed in the image. The provided equation is:
\[ 2y' = \sqrt{x + 2y + 1} \]
### Explanation of Terms:
- \( y' \) denotes the first derivative of \( y \) with respect to \( x \), commonly referred to as the rate of change of \( y \) concerning \( x \).
- The square root symbol \(\sqrt{ }\) indicates the principal square root of the expression within it.
- The expression inside the square root, \( x + 2y + 1 \), involves the independent variable \( x \) and the dependent variable \( y \).
### Steps for Solving the Equation:
1. **Isolate the Derivative Term:**
To simplify the analysis, we start by rearranging the given equation to express \( y' \) explicitly.
\[ y' = \frac{1}{2} \sqrt{x + 2y + 1} \]
2. **Identify Methods for Solving:**
- This is a first-order ordinary differential equation (ODE).
- Depending on the context, methods such as separation of variables, integrating factor, or numerical methods might be applied to solve the equation.
3. **Application and Interpretation:**
- Differential equations like this can model various physical phenomena such as growth rates, decay processes, or motion of objects.
- Understanding the solution of this differential equation can provide insights into the relationship between \( x \) and \( y \) and how changes in one affect the other.
By systematically addressing steps like isolating the derivative term and exploring suitable solving techniques, one can find the functional relationship between \( x \) and \( y \) that satisfies this differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b9e8ebd-ceba-4fff-be05-4ffc155e08e3%2Fb6e1d3f7-6b61-432b-ba8d-acd4f2559054%2Fr8m1s_processed.png&w=3840&q=75)
Transcribed Image Text:In this educational segment, we will discuss the differential equation displayed in the image. The provided equation is:
\[ 2y' = \sqrt{x + 2y + 1} \]
### Explanation of Terms:
- \( y' \) denotes the first derivative of \( y \) with respect to \( x \), commonly referred to as the rate of change of \( y \) concerning \( x \).
- The square root symbol \(\sqrt{ }\) indicates the principal square root of the expression within it.
- The expression inside the square root, \( x + 2y + 1 \), involves the independent variable \( x \) and the dependent variable \( y \).
### Steps for Solving the Equation:
1. **Isolate the Derivative Term:**
To simplify the analysis, we start by rearranging the given equation to express \( y' \) explicitly.
\[ y' = \frac{1}{2} \sqrt{x + 2y + 1} \]
2. **Identify Methods for Solving:**
- This is a first-order ordinary differential equation (ODE).
- Depending on the context, methods such as separation of variables, integrating factor, or numerical methods might be applied to solve the equation.
3. **Application and Interpretation:**
- Differential equations like this can model various physical phenomena such as growth rates, decay processes, or motion of objects.
- Understanding the solution of this differential equation can provide insights into the relationship between \( x \) and \( y \) and how changes in one affect the other.
By systematically addressing steps like isolating the derivative term and exploring suitable solving techniques, one can find the functional relationship between \( x \) and \( y \) that satisfies this differential equation.
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