Problem 36 - Calculus Concepts

Question

Write a differential equation expressing the information given and, when possible, find a general solution for the differential equation. The rate of growth of the height \(h\) of a young child with respect to the age \(y\) of the child decreases in inverse proportion to the age of the child.

Step-by-Step Solution

Answer

The differential equation is \( \frac{dh}{dy} = \frac{k}{y} \), and its general solution is \( h = k \ln|y| + C \).

Step 1: Identify the relationship

The problem states that the rate of growth of the child's height, \( \frac{dh}{dy} \), is inversely proportional to the age \( y \). This relationship can be expressed mathematically as \( \frac{dh}{dy} = \frac{k}{y} \), where \( k \) is a constant of proportionality.

Step 2: Write the differential equation

Based on the inverse proportional relationship, the differential equation is \( \frac{dh}{dy} = \frac{k}{y} \).

Step 3: Separate variables

To solve the differential equation, we start by separating the variables. Multiply both sides by \( dy \) to get \( dh = \frac{k}{y} dy \).

Step 4: Integrate both sides

Integrate both sides of the equation \( \int dh = \int \frac{k}{y} dy \). This results in \( h = k \ln|y| + C \), where \( C \) is the constant of integration.

Step 5: State the general solution

The general solution to the differential equation is \( h = k \ln|y| + C \). This expression describes how the height of the child \( h \) changes with age \( y \).

Key Concepts

Rate of Growth

In differential equations, the rate of growth refers to how a quantity changes with respect to another variable over time. In this case, the rate of growth is expressed as how the height \( h \) of a child changes with respect to age \( y \). Mathematically, this relationship is depicted using the derivative \( \frac{dh}{dy} \), which signifies the change in height per small increment of age.
This rate of growth is not constant; instead, it varies depending on other factors, such as the child's age. Understanding how quickly a child grows can help in predicting future development milestones. By capturing this information through the rate of growth, differential equations can offer insights into patterns over time. It allows us to make predictions and understand the dynamics of growth processes in biology and other fields.

Inverse Proportionality

Inverse proportionality is a relationship between two variables where as one increases, the other decreases at a proportional rate. In the child's growth problem, the rate of growth of height \( \frac{dh}{dy} \) is inversely proportional to the child's age \( y \).
This means that as the child ages, the growth rate slows down. Mathematically, this can be expressed as \( \frac{dh}{dy} = \frac{k}{y} \). Here, \( k \) is a constant, indicating that the product of the rate of growth and the age remains steady. Understanding inverse proportionality helps in comprehending relationships where growth diminishes as a factor like age increases, a common scenario in natural phenomena.

Integration

Integration is a fundamental technique used to solve differential equations. It allows us to find a function from its derivative, essentially reversing differentiation. In the provided problem, after separating variables, integration is used to solve \( \int dh = \int \frac{k}{y} dy \).
This process involves integrating each side of the equation independently. The left side yields \( h \) since the derivative of \( h \) with respect to \( y \) was initially given. For the right side, integrating \( \frac{k}{y} \) results in \( k \ln|y| \), a function representing the height in terms of age.

• Integration helps in finding the original function that describes the system.
• It is an essential mathematical tool for obtaining general solutions of differential equations.

Ultimately, integration simplifies the understanding of how quantities in a system interrelate over time.

General Solution

A general solution to a differential equation encompasses all possible specific solutions. In the context of the problem, the general solution \( h = k \ln|y| + C \) represents a family of curves that describe how the child's height evolves as they age.
Each specific solution within this family corresponds to a particular value of the constant of integration, \( C \). This constant accounts for initial conditions or specific data points that may personalize the solution to a given scenario.
Understanding general solutions is crucial for gaining a complete picture of differential equations, as they provide a comprehensive framework that can be adjusted to fit particular circumstances. This flexibility is essential in modeling real-world situations where initial conditions or constraints often vary across different cases.