Use the method for solving Bernoulli equations to solve the following differential equation. 1 3 dy y x-5 + -= 4(x - 5)y dx Find an expression for v in the form v=y¹-n. V=
Use the method for solving Bernoulli equations to solve the following differential equation. 1 3 dy y x-5 + -= 4(x - 5)y dx Find an expression for v in the form v=y¹-n. V=
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 Bernoulli Differential Equations
To solve the differential equation using the method for Bernoulli equations, we start with the given equation:
\[ \frac{dy}{dx} + \frac{y}{x - 5} = 4(x - 5)y^{\frac{1}{3}} \]
#### Step-by-Step Solution
1. First, identify the Bernoulli differential equation form:
\[ \frac{dy}{dx} + P(x)y = Q(x)y^n \]
In our equation, \( P(x) = \frac{1}{x-5} \), \( Q(x) = 4(x-5) \), and \( n = \frac{1}{3} \).
2. Transform the equation using the substitution \( v = y^{1-n} \):
For our equation (\( n = \frac{1}{3} \)), we get:
\[
v = y^{1 - \frac{1}{3}} = y^{\frac{2}{3}}
\]
Therefore, \( v = y^{\frac{2}{3}} \).
---
Place this substituted expression ( \( v = y^{\frac{2}{3}} \) ) in the substitution box:
\[ v = \]
---
This substitution simplifies the differential equation and allows it to be solved more easily using standard techniques for linear differential equations. Continue with further steps to find the explicit solution if required.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe57d9078-3a26-4ab9-a5c6-f67f173c178d%2Fae174a47-8c58-4fc6-b6aa-2df9c714be43%2Fokgd7b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Bernoulli Differential Equations
To solve the differential equation using the method for Bernoulli equations, we start with the given equation:
\[ \frac{dy}{dx} + \frac{y}{x - 5} = 4(x - 5)y^{\frac{1}{3}} \]
#### Step-by-Step Solution
1. First, identify the Bernoulli differential equation form:
\[ \frac{dy}{dx} + P(x)y = Q(x)y^n \]
In our equation, \( P(x) = \frac{1}{x-5} \), \( Q(x) = 4(x-5) \), and \( n = \frac{1}{3} \).
2. Transform the equation using the substitution \( v = y^{1-n} \):
For our equation (\( n = \frac{1}{3} \)), we get:
\[
v = y^{1 - \frac{1}{3}} = y^{\frac{2}{3}}
\]
Therefore, \( v = y^{\frac{2}{3}} \).
---
Place this substituted expression ( \( v = y^{\frac{2}{3}} \) ) in the substitution box:
\[ v = \]
---
This substitution simplifies the differential equation and allows it to be solved more easily using standard techniques for linear differential equations. Continue with further steps to find the explicit solution if required.
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