Problem 81 - Calculus Single Variable

Question

Let \(x\) and \(y\) be the measures of two body parts with relative growth rates that are proportional to a common factor \(\Phi(t)\) $$ \frac{1}{x} \cdot \frac{d x}{d t}=\alpha \cdot \Phi(t) \quad \text { and } \quad \frac{1}{y} \cdot \frac{d y}{d t}=\beta \cdot \Phi(t) $$ Show that \(x\) and \(y\) satisfy the Huxley Allometry Equation \(y=k x^{p}\) for suitable constants \(k\) and \(p\)

Step-by-Step Solution

Answer

The growth equations show that \( x \) and \( y \) satisfy \( y = k x^p \) where \( k = \exp(C_2) \) and \( p = \frac{\beta}{\alpha} \).

Step 1: Understanding the Differential Equations

The given equations describe the relative growth rates of two body parts, \( x \) and \( y \), with respect to time \( t \). These are first-order differential equations: \( \frac{1}{x} \cdot \frac{d x}{d t} = \alpha \cdot \Phi(t) \) and \( \frac{1}{y} \cdot \frac{d y}{d t} = \beta \cdot \Phi(t) \). The growth rates are proportional to some common factor \( \Phi(t) \).

Step 2: Separate Variables and Integrate

Rewriting the equations, we have: \( \frac{d x}{x} = \alpha \Phi(t) \, d t \) and \( \frac{d y}{y} = \beta \Phi(t) \, d t \). We can integrate both sides to find expressions for \( x \) and \( y \).

Step 3: Integrate the Equation for \( x \)

Integrating, \( \int \frac{d x}{x} = \int \alpha \Phi(t) \, d t \). This simplifies to \( \ln|x| = \alpha \int \Phi(t) \, d t + C_1 \), where \( C_1 \) is the integration constant.

Step 4: Integrate the Equation for \( y \)

Similarly, integrate the equation for \( y \): \( \int \frac{d y}{y} = \int \beta \Phi(t) \, d t \). This gives \( \ln|y| = \beta \int \Phi(t) \, d t + C_2 \), where \( C_2 \) is another constant of integration.

Step 5: Eliminate \( \Phi(t) \)

From the two integrated expressions, set the expressions \( \alpha \int \Phi(t)\,d t + C_1 \) and \( \beta \int \Phi(t)\,d t + C_2 \) in terms of each other: \( \ln|y| - C_2 = \frac{\beta}{\alpha} (\ln|x| - C_1) \).

Step 6: Express \( y \) in Terms of \( x \)

Simplify to find \( y = \exp(C_2) \cdot \exp \left( \frac{\beta}{\alpha} \ln|x| \right) \). This simplifies to \( y = k x^p \) where \( k = \exp(C_2) \) and \( p = \frac{\beta}{\alpha} \).

Step 7: Conclusion

Thus, the derived equation \( y = k x^p \) confirms that \( x \) and \( y \) satisfy the Huxley Allometry Equation for suitable constants \( k \) and \( p \).

Key Concepts

Differential Equations

A differential equation is an equation that involves an unknown function and its derivatives. In the context of this exercise, we are examining the relative growth rates of two body parts, which means we have two separate differential equations:

• For the body part represented by variable \( x \), the equation is \( \frac{1}{x} \cdot \frac{d x}{d t} = \alpha \cdot \Phi(t) \).
• For the body part represented by variable \( y \), the equation is \( \frac{1}{y} \cdot \frac{d y}{d t} = \beta \cdot \Phi(t) \).

These equations indicate that the growth of each body part is proportional to a common growth factor, \( \Phi(t) \), through constants \( \alpha \) and \( \beta \). The task is to show that these equations can result in a specific relationship known as the Huxley Allometry Equation. Understanding the setup of these equations is crucial as a first step in the solution process. Each equation is first-order, meaning it involves only the first derivative of the function. This level of complexity is typical in many biological and physical growth models.
By identifying the right techniques, such as separation of variables and integration, we can solve these equations and uncover the relationship between \( x \) and \( y \).

Relative Growth Rates

Relative growth rates refer to the rate of change of a quantity relative to the quantity itself. In our equations, it's represented as \( \frac{1}{x} \cdot \frac{d x}{d t} \) for the variable \( x \) and \( \frac{1}{y} \cdot \frac{d y}{d t} \) for the variable \( y \).

• This means we are observing how fast something grows in relation to its current size.
• Expressing growth this way allows us to measure how each body part grows relative to its current size at any given time \( t \).

In biology, this type of measure is particularly useful for comparative growth studies, as it helps in understanding ratios and proportions of growth between organisms or their parts.
Due to the factor \( \Phi(t) \), which is common to both equations, these growth rates are tied together. This commonality is what eventually allows us to find a relationship between \( x \) and \( y \). The key is realizing that the rates are not independent, which is a typical finding in many biological systems where growth is influenced by shared environmental or internal factors.

Variable Separation

The method of variable separation is used to simplify and solve differential equations. This technique involves rearranging the equation so that each variable and its differential are on opposite sides of the equation.
For our problem, we begin with:

• For \( x \): \( \frac{d x}{x} = \alpha \Phi(t) \, d t \)
• For \( y \): \( \frac{d y}{y} = \beta \Phi(t) \, d t \)

By separating the variables this way, we can now integrate each side independently. This method is powerful because it transforms a complex relationship into a form that can be addressed using integration.
This step is key to solving our differential equations and discovering how the variables \( x \) and \( y \) interrelate over time. Proper separation sets up the pathway for the subsequent step—integration—which allows us to solve for \( x \) and \( y \) as explicit functions.

Integration

Integration is the mathematical process of finding a function given its derivative. In solving differential equations, integration helps us find explicit forms of the relationships we study. Once we have separated our variables for both \( x \) and \( y \):

• We integrate \( \int \frac{d x}{x} = \int \alpha \Phi(t) \, d t \).
• This gives us \( \ln|x| = \alpha \int \Phi(t) \, d t + C_1 \), where \( C_1 \) is a constant of integration.
• Similarly for \( y \), we integrate \( \int \frac{d y}{y} = \int \beta \Phi(t) \, d t \).
• This results in \( \ln|y| = \beta \int \Phi(t) \, d t + C_2 \), where \( C_2 \) is another constant.

These solutions express the logarithmic relationship between the size of the body parts and their growth over time, determined by \( \Phi(t) \). Solutions often include constants because integration is an indefinite process without specific boundary or initial conditions.
From these integrated equations, by manipulating their forms, we discover an algebraic relationship expressed in the Huxley Allometry Equation: \( y = k x^p \). The values for \( k \) and \( p \) come from the constants in the integrated form, showing how specific initial conditions and constant factors influence the growth pattern.