Use the method for solving homogeneous equations to solve the following differential equation. dy de 70 sec 0 +y
Use the method for solving homogeneous equations to solve the following differential equation. dy de 70 sec 0 +y
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 17E: Find the general solution for each differential equation. Verify that each solution satisfies the...
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![### Solving Homogeneous Equations: Differential Equation Problem
**Problem Statement:**
Use the method for solving homogeneous equations to solve the following differential equation.
\[
\frac{dy}{d\theta} = \frac{70\theta \sec\left(\frac{y}{\theta}\right)}{\theta} + y
\]
---
**Instructions:**
Ignoring lost solutions, if any, the general solution is \( y = \text{[Input Box]} \).
*(Type an expression using \(\theta\) as the variable.)*
---
**Explanation:**
**Given Differential Equation:**
\[
\frac{dy}{d\theta} = 70\sec\left( \frac{y}{\theta} \right) + y
\]
This differential equation can be classified and solved using methods suitable for homogeneous equations. A homogeneous equation is one where you can rewrite it such that the dependent and independent variables are separable via substitution. In this context, typically, you would use the substitution \( v = \frac{y}{\theta} \).
Here is a step-by-step outline for solving this type of differential equation:
1. **Substitution:**
Define \( v = \frac{y}{\theta} \), hence \( y = v\theta \).
2. **Differentiation:**
Differentiate \( y = v\theta \) with respect to \(\theta\):
\[
\frac{dy}{d\theta} = v + \theta\frac{dv}{d\theta}
\]
3. **Substitute and Simplify:**
Substitute \( y \) and \( \frac{dy}{d\theta} \) back into the original equation, and solve for \( v \).
4. **Solve the Resulting Equation:**
Solve the resulting ordinary differential equation to find \( v \) in terms of \(\theta\).
5. **Back-substitution:**
Finally, revert the substitution \( v = \frac{y}{\theta} \) to express \( y \) in terms of \(\theta\).
These steps will get you from the given homogeneous differential equation to its solution. Remember to check for any potential solutions that may have been lost during this process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1887966a-d451-4216-a13a-dc9720678597%2Fc1da5bad-55a6-423a-93e7-826b5d08890b%2Fq3n30lm_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Homogeneous Equations: Differential Equation Problem
**Problem Statement:**
Use the method for solving homogeneous equations to solve the following differential equation.
\[
\frac{dy}{d\theta} = \frac{70\theta \sec\left(\frac{y}{\theta}\right)}{\theta} + y
\]
---
**Instructions:**
Ignoring lost solutions, if any, the general solution is \( y = \text{[Input Box]} \).
*(Type an expression using \(\theta\) as the variable.)*
---
**Explanation:**
**Given Differential Equation:**
\[
\frac{dy}{d\theta} = 70\sec\left( \frac{y}{\theta} \right) + y
\]
This differential equation can be classified and solved using methods suitable for homogeneous equations. A homogeneous equation is one where you can rewrite it such that the dependent and independent variables are separable via substitution. In this context, typically, you would use the substitution \( v = \frac{y}{\theta} \).
Here is a step-by-step outline for solving this type of differential equation:
1. **Substitution:**
Define \( v = \frac{y}{\theta} \), hence \( y = v\theta \).
2. **Differentiation:**
Differentiate \( y = v\theta \) with respect to \(\theta\):
\[
\frac{dy}{d\theta} = v + \theta\frac{dv}{d\theta}
\]
3. **Substitute and Simplify:**
Substitute \( y \) and \( \frac{dy}{d\theta} \) back into the original equation, and solve for \( v \).
4. **Solve the Resulting Equation:**
Solve the resulting ordinary differential equation to find \( v \) in terms of \(\theta\).
5. **Back-substitution:**
Finally, revert the substitution \( v = \frac{y}{\theta} \) to express \( y \) in terms of \(\theta\).
These steps will get you from the given homogeneous differential equation to its solution. Remember to check for any potential solutions that may have been lost during this process.
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