Problem 3 - Calculus for Biology and Medicine

Question

\text { In Problems , solve each pure-time differential equation. } $$ \frac{d y}{d x}=\frac{1}{x}, \text { where } y(1)=0 . $$

Step-by-Step Solution

Answer

The solution is \( y = \ln |x| \).

Step 1 - Identify the Type of Differential Equation

The given differential equation is \( \frac{dy}{dx} = \frac{1}{x} \). This is a first-order pure-time differential equation.

Step 2 - Integrate Both Sides

To solve the differential equation, integrate both sides with respect to \( x \). This gives:\[\int \frac{dy}{dx} \, dx = \int \frac{1}{x} \, dx\] which simplifies to \(y = \ln |x| + C\), where \( C \) is the integration constant.

Step 3 - Apply the Initial Condition

Use the initial condition \( y(1) = 0 \) to find the constant \( C \). Substitute \( x = 1 \) and \( y = 0 \) into the solution:\[0 = \ln |1| + C\]Since \( \ln |1| = 0 \), we have \( C = 0 \).

Step 4 - Write the Final Solution

Substitute the value of \( C \) back into the integrated equation:\[y = \ln |x| + 0\]Thus, the solution to the differential equation is \( y = \ln |x| \).

Key Concepts

First-Order Differential Equations

A first-order differential equation involves derivatives of a function dependent on one variable. In our exercise, the equation \( \frac{dy}{dx} = \frac{1}{x} \) is first-order because it contains the first derivative \( \frac{dy}{dx} \). This type of equation focuses on the relationship between a function and its rate of change.

Such differential equations can be expressed in the form \( \frac{dy}{dx} = f(x) \), where \( f(x) \) is some function of \( x \).
Understanding first-order equations is key because they model various everyday phenomena:

• Population growth
• Radioactive decay
• Cooling or heating over time

Knowing how to identify and solve first-order differential equations is a fundamental skill in calculus and is applicable in numerous scientific fields.

Integration

Integration is a powerful tool in mathematics that helps in finding the original function from its derivative. In this exercise, integration is crucial for solving the differential equation \( \frac{dy}{dx} = \frac{1}{x} \).

By integrating both sides with respect to \( x \), the left side transforms into \( y \) and the right side becomes \( \int \frac{1}{x} \, dx \).
This integral simplifies to the natural logarithm expression: \( y = \ln |x| + C \), where \( C \) is an arbitrary constant.

It's important to understand the process and:

• Recognize the integral of basic functions
• Apply rules such as integration by parts or substitution
• Manipulate expressions to bring them into an integrable form

Integration helps bridge the gap in understanding how functions behave and evolve, especially over time or other domains.

Initial Conditions

Initial conditions are specific values that allow you to determine the constants involved in the general solution of differential equations. In our case, the initial condition \( y(1) = 0 \) is applied after integrating the differential equation.

This step is where you substitute the initial values into the general solution. By doing this:

• We establish the particular solution by fixing the constant \( C \)
• For \( x = 1 \) and \( y = 0 \), substitute into \( y = \ln |x| + C \)
• This gave \( C = 0 \) as \( \ln |1| = 0 \)

Using initial conditions makes the solution unique and tailored to a specific scenario, unlike the general solution which still contains arbitrary constants.