Problem 78 - Calculus for Biology and Medicine

Question

Solve by rewriting the differential equation as an equation for \(\frac{d x}{d y}\) : $$ \frac{d y}{d x}=\frac{y}{y+1}, \text { for } x \geq 0 \text { with } y(0)=1 $$

Step-by-Step Solution

Answer

The equation for \( \frac{d x}{d y} \) is \( \frac{d x}{d y} = 1 + \frac{1}{y} \).

Step 1: Understanding the Problem

The given differential equation is \( \frac{d y}{d x} = \frac{y}{y+1} \). We need to rewrite it as an equation for \( \frac{d x}{d y} \).

Step 2: Finding the Reciprocal

To rewrite the equation for \( \frac{d x}{d y} \), we need to take the reciprocal of \( \frac{d y}{d x} \). Thus, \( \frac{d x}{d y} = \frac{1}{\frac{d y}{d x}} = \frac{y+1}{y} \).

Step 3: Simplification

The expression found for \( \frac{d x}{d y} \) simplifies directly to \( \frac{d x}{d y} = 1 + \frac{1}{y} \).

Key Concepts

Understanding Calculus in Differential Equations

Calculus is a branch of mathematics that deals with how things change and lets us understand the behavior of functions and their derivatives. In differential equations, we use calculus to find unknown functions. A differential equation involves functions and their rates of change, represented by derivatives.

In our given problem, we have a first-order differential equation. It shows the rate of change of a function, in this case, representing how "y" changes with respect to "x". The derivative is written as \(\frac{d y}{d x}\), which means it's the derivative of “y” with respect to “x”.

The task here involves switching this perspective to focus on how "x" changes with respect to "y". This requires a solid understanding of how derivatives can be manipulated, which is a fundamental part of calculus. By understanding how these derivatives are interconnected, we can translate problems into forms that are easier to solve or gain insight from.

The Concept of Reciprocals

Reciprocals are an essential mathematical tool used in a variety of contexts, including calculus. A reciprocal is essentially flipping a fraction. For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). This is well-known from operations involving division of fractions.

In the context of differential equations, finding the reciprocal changes the relationship between the variables. Initially, we had \(\frac{d y}{d x} = \frac{y}{y+1}\). To find how "x" changes in relation to "y", the reciprocal gives us \(\frac{d x}{d y} = \frac{1}{\frac{d y}{d x}}\). Switching the focus in this way allows us to view the problem from another angle.

Using reciprocals is a simple yet powerful mathematical maneuver that helps solve equations in a more straightforward manner, adapting the mathematics to fit the problem at hand.

Simplification for Clearer Solutions

Simplification is the process of making an expression easier to work with. In mathematics, this often involves reducing fractions, combining like terms, or eliminating unnecessary complexity while preserving the essential features of the expression.

In our differential equation reformulation, after finding \(\frac{d x}{d y} = \frac{y+1}{y}\), further simplification reveals \(\frac{d x}{d y} = 1 + \frac{1}{y}\). This form is often easier to analyze or integrate, providing a clearer path to a solution.

Simplifying differential equations not only helps in solving them more efficiently, but it also enhances understanding by translating mathematical complexities into more manageable forms. This allows for greater focus on the underlying behavior of the functions involved.