Problem 76 - Calculus Single Variable

Question

Suppose that \(\lambda, A,\) and \(p\) are positive constants. Solve the differential equation $$ \frac{d y}{d t}=\frac{\lambda}{p} \cdot t^{p-1} \cdot(A-y) $$ A solution of this equation is known as a Janoschek growth function. Such functions are often used to model animal growth. The parameter \(A\) is called "the size of the adult." Why?

Step-by-Step Solution

Answer

The parameter \(A\) represents the adult size because as \(t\rightarrow\infty\), \(y\) approaches \(A\).

Step 1: Identify Separable Variables

The given differential equation is \( \frac{d y}{d t}=\frac{\lambda}{p} \cdot t^{p-1} \cdot(A-y) \). Identify if it is a separable differential equation.Recognize that separable differential equations have the form \( g(y)\,dy = f(t)\,dt \). In this case, separate the variables \( y \) and \( t \):\[ \frac{dy}{A-y} = \frac{\lambda}{p} \cdot t^{p-1} \, dt \].

Step 2: Integrate Both Sides

Integrate both sides of the separated equation:For the left side, integrate with respect to \( y \):\[ \int \frac{1}{A-y} \, dy = -\ln|A-y| + C_1 \]For the right side, integrate with respect to \( t \):\[ \int \frac{\lambda}{p} \cdot t^{p-1} \, dt = \frac{\lambda}{p^2} \cdot t^p + C_2 \].

Step 3: Combine and Simplify the Expression

After integration, equate the two expressions with their respective constants:\[ -\ln|A-y| = \frac{\lambda}{p^2} t^p + C \]where \( C = C_2 - C_1 \). Simplify by removing the negative sign:\[ \ln|A-y| = -\frac{\lambda}{p^2} t^p - C \].

Step 4: Solve for \( y \)

Exponentiate both sides to solve for \( y \):\[ |A-y| = e^{-\frac{\lambda}{p^2} t^p - C} \]Since \( y < A \), we have:\[ A-y = e^{-C} e^{-\frac{\lambda}{p^2} t^p} \].

Step 5: Express Final Solution for \( y \)

Rewrite \( y \) in terms of the constants and variables:\[ y = A - Ce^{-\frac{\lambda}{p^2} t^p} \].This function growths towards \( A \) as \( t \to \infty \), since the exponential term tends to zero.

Key Concepts

Janoschek growth function

The Janoschek growth function is a mathematical representation often used in the realm of biological growth modeling. It describes how an entity grows over time, particularly useful in understanding the growth patterns of living organisms. Such a function is characterized by parameters that determine its specific behavior.
For this function, the growth rate is not constant; it changes with time, depending on how far the current size is from the ultimate size or capacity (denoted by \( A \)). The parameter \( A \), often called "the size of the adult," represents the asymptotic limit the growth will approach as time continues. This means that no matter what the initial conditions are, the function will evolve in such a way that the value of \( y \), representing size, approaches \( A \) as \( t \to \infty \).
The Janoschek function is significant because it provides a more realistic model for many biological processes, where growth slows down as the organism approaches its ultimate size. It helps researchers predict future growth based on current data, which is especially useful in fields such as ecology and agriculture.

Separable variables

Differential equations can sometimes be simplified using the method of separable variables, a technique that breaks down the equation by isolating each variable on different sides of the equation. This method allows us to simplify solving differential equations by making integration possible on a term-by-term basis.
In the exercise, the differential equation is expressed as \( \frac{d y}{d t} = \frac{\lambda}{p} \cdot t^{p-1} \cdot (A-y) \). To separate this equation, we move the terms involving \( y \) on one side and the terms involving \( t \) on the other:

• Divide both sides by \( A-y \) to isolate the \( y \)-terms:
• \( \frac{dy}{A-y} = \frac{\lambda}{p} \cdot t^{p-1} \, dt \)

This process transforms the problem into simpler integrals for each variable, making it easier to integrate and find a solution. Recognizing separable variables is crucial, as it transforms a complex equation into one that can be systematically solved through integration.

Animal growth modeling

Animal growth modeling utilizes mathematical equations to predict and understand the way animals grow over time. Models, like the Janoschek growth function, are particularly useful because they can encapsulate complex biological principles in an equation that predicts growth.
Growth models are essential for various reasons:

• They help in maximizing livestock productivity by predicting how animals will grow under specific conditions.
• Models assist conservation efforts by understanding how various species might grow in changing environments.
• In research, these models provide insight into the biological limits of growth and the factors affecting those limits.

The Janoschek function, characterized by its asymptotic behavior towards an ultimate size, reflects the natural slowing of growth as animals near their mature stage. This characteristic aligns well with observed animal growth patterns, making such models both practical and theoretically robust for scientific analysis.

Integration technique

The integration technique is a crucial mathematical process used to solve differential equations, such as the one in this exercise. Integration allows us to find a function when its rate of change is known, which is essential for solving equations involving growth models.
In the step-by-step solution, after separating variables, we faced two integrals that needed solving:

• For the left side, integrating \( \int \frac{1}{A-y} \, dy \) results in \( -\ln|A-y| \). This involves understanding the natural logarithm as the antiderivative of reciprocal functions.
• For the right side, integrating \( \int \frac{\lambda}{p} \cdot t^{p-1} \, dt \) results in \( \frac{\lambda}{p^2} \cdot t^p \), where power rule of integration is applied.

After integrating, converting these results into an expressive form is key. By exponentiating both sides of the simplified equation, we solve for \( y \), uncovering a function that models the growth described by the original differential equation.
Understanding these integration techniques is fundamental for anyone studying calculus or applying mathematical models to biological phenomena, as integration reveals the behavior of dynamic systems over time.