Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 19
Solve the Logistic equation, \(d y / d t=\alpha y(1-\) \((1 / K) y\) ), by viewing it as a Bernoulli equation and then solve the resulting linear equation by using an integrating factor rather than the method of undetermined coefficients that is illustrated in the examples.
Short Answer
Expert verified
By viewing it as a Bernoulli equation and using an integrating factor, the solution is \( y = \frac{Ke^{\alpha t}}{C + e^{\alpha t}} \).
Step by step solution
01
Identify the Bernoulli equation form
Recognize the given logistic equation \(\frac{dy}{dt} = \alpha y(1 - \frac{y}{K})\). Rewrite it in the form of a Bernoulli equation \(\frac{dy}{dt} + P(t)y = Q(t) y^n\) with \ n = 2 \.
02
Divide through by y
To transform the equation, divide through by y: \(\frac{1}{y} \frac{dy}{dt} = \alpha (1 - \frac{y}{K})\), simplifying to \( \frac{dy}{dt} + \-\frac{\alpha}{K} y = \alpha y \).
03
Set up the standard Bernoulli form
Rewrite to fit the Bernoulli form. In this case: \( \frac{dy}{dt} + \frac{\alpha}{K} y = \alpha \).
04
Use the substitution
Let \( v = y^{1 - 1} = y^0 \), simplifying \( \frac{dy}{dt} + \frac{\alpha}{K} y = \alpha y\) to a linear form.
05
Differentiate the substitution
Differentiate \( v = y^{0} \) with respect to \( t \) to get \(\frac{dv}{dt}\frac{dy}{dt} + \frac{\alpha}{K} \frac{dy}{dt} \).
06
Simplify the linear equation
The substitution transforms the equation into a linear one: \( \frac{dv}{dt} + \frac{\alpha}{K} v = \frac{\alpha}{K} \).
07
Find the integrating factor
Identify the integrating factor \( \mu(t) = e^{\frac{\alpha}{K}t} \).
08
Multiply through by the integrating factor
Multiply each term of the differential equation by \( \mu(t) \) to facilitate integration: \( e^{\frac{\alpha}{K}t} \frac{dv}{dt} + \frac{\alpha}{K} e^{\frac{\alpha}{K}t} v = \frac{\alpha}{K} e^{\frac{\alpha}{K} t} \).
09
Integrate both sides
Integrate the simplified equation with respect to \( t \: \frac{d}{dt} (e^{\frac{\alpha}{K} t} v = \frac{\alpha}{K} e^{\frac{\alpha}{K} t} \rightarrow e^{\frac{\alpha}{K} t} v = \alpha t + C\).
10
Solve for the original variable
Solve the resulting equation for \( v \) first, then substitute back to \( y \: y = \frac{Ke^{\alpha t}}{C + e^{\alpha t}}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Bernoulli Equation
The Bernoulli Equation is a special type of non-linear differential equation of the form \( \frac{dy}{dt} + P(t)y = Q(t)y^n \). It鈥檚 named after Jacob Bernoulli, a Swiss mathematician.
Understanding how to manipulate non-linear equations into a more manageable form is crucial. Substitution helps convert it into a linear differential equation, which is easier to solve using standard methods.
- When solving a Bernoulli equation, the first step is often to identify how it can be manipulated into a standard linear form. This usually involves substitution and transformation.
- For our Logistic Equation problem, the equation is rewritten to fit this form. Initially, we鈥檙e given \( \frac{dy}{dt} = \alpha y(1 - \frac{y}{K}) \). To match the Bernoulli form, we divide through to simplify and identify the parameters, which helps us proceed with solving the equation.
- The critical step to transforming it is to use the substitution \( v = y^{1 - n} \), which results in a linear differential equation.
Understanding how to manipulate non-linear equations into a more manageable form is crucial. Substitution helps convert it into a linear differential equation, which is easier to solve using standard methods.
Integrating Factor
An integrating factor is used to solve linear differential equations.
Let's see how it applies:
In our example, after simplifying the Bernoulli equation to \( \frac{dv}{dt} + \frac{\alpha}{K} v = \frac{\alpha}{K} \), we identify an integrating factor as \( \mu(t) = e^{\frac{\alpha}{K} t} \).
This leads to solving \( \frac{d}{dt} (e^{\frac{\alpha}{K} t} v) = \frac{\alpha}{K} e^{\frac{\alpha}{K} t} \) by integrating both sides, thus easing the computation considerably.
- The method involves multiplying through the differential equation by a specific function, called the integrating factor, denoted as \( \mu(t) \), which simplifies the equation such that it can be integrated directly.
Let's see how it applies:
In our example, after simplifying the Bernoulli equation to \( \frac{dv}{dt} + \frac{\alpha}{K} v = \frac{\alpha}{K} \), we identify an integrating factor as \( \mu(t) = e^{\frac{\alpha}{K} t} \).
- This integrating factor transforms our differential equation into a form that's directly integrable.
- Essentially, it helps to accumulate the effects of the changing variable. Multiplying both sides by this factor aligns the differential equation to a product rule form, making integration straightforward.
This leads to solving \( \frac{d}{dt} (e^{\frac{\alpha}{K} t} v) = \frac{\alpha}{K} e^{\frac{\alpha}{K} t} \) by integrating both sides, thus easing the computation considerably.
Differential Equations Solving Methods
Solving differential equations involves various methods, including separation of variables, integrating factors, and substitutions.
Each method has its specific use cases and becomes handy depending on the nature of the differential equation.
In our logistic equation example, the process combines recognizing the equation form, transforming using substitution, and then applying integrating factors to find the solution efficiently.
- Separation of Variables: This is one of the most basic methods where you rearrange the differential equation so that each variable appears on a different side of the equation. Then integrate both sides with respect to their respective variables.
- Integrating Factor: As described, it's an effective method for solving linear first-order differential equations by multiplying the equation by a strategically chosen function.
- Bernoulli Equation Transformation: For non-linear equations in the Bernoulli form, a substitution is used to transform them into a linear equation that can then be solved by other standard methods like integrating factors.
Each method has its specific use cases and becomes handy depending on the nature of the differential equation.
In our logistic equation example, the process combines recognizing the equation form, transforming using substitution, and then applying integrating factors to find the solution efficiently.
