Problem 55 - Calculus for Biology and Medicine

Question

In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d s}=\sin (\pi s), 0 \leq s \leq 1 $$

Step-by-Step Solution

Answer

The general solution is \( y(s) = -\frac{1}{\pi} \cos(\pi s) + C \).

Step 1: Interpret the Differential Equation

The given differential equation is \( \frac{dy}{ds} = \sin(\pi s) \). This means that the derivative of \( y \) with respect to \( s \) is equal to \( \sin(\pi s) \). Our task is to find the function \( y(s) \) that satisfies this equation.

Step 2: Integrate Both Sides

To find \( y(s) \), we need to integrate both sides of the equation with respect to \( s \). The equation becomes:\[ y(s) = \int \sin(\pi s) \, ds \]

Step 3: Determine the Integral

The integral of \( \sin(\pi s) \) with respect to \( s \) is found using a substitution method or known integral formulas. The result is: \[ y(s) = -\frac{1}{\pi}\cos(\pi s) + C \]where \( C \) is the integration constant.

Step 4: Write the General Solution

The general solution to the differential equation \( \frac{dy}{ds} = \sin(\pi s) \) is:\[ y(s) = -\frac{1}{\pi}\cos(\pi s) + C \]This includes an arbitrary constant \( C \), which accounts for the family of possible solutions.

Key Concepts

The Art of Integration

Integration is a core part of calculus that involves finding the antiderivative or the integral of a function. When you encounter a differential equation like \( \frac{d y}{d s} = \sin(\pi s) \), integration helps you reverse the process of differentiation.

To integrate, you look for a function \( y(s) \) whose derivative is \( \sin(\pi s) \). In this case, integrating \( \sin(\pi s) \) involves recognizing patterns or using substitution methods.

• Recall that the derivative of \( -\cos(x) \) is \( \sin(x) \). Apply this to \( \sin(\pi s) \).
• Use a constant factor to adjust for \( \pi \), resulting in \( y(s) = -\frac{1}{\pi}\cos(\pi s) + C \).

The integration constant \( C \) arises naturally in indefinite integrals because integration is essentially undoing differentiation, leaving room for a constant addition.
Understanding integration is crucial for solving differential equations, as it bridges the step from a derivative back to the original function, providing a fundamental tool in calculus, physics, and engineering.

Understanding Trigonometric Functions in Calculus

Trigonometric functions like \( \sin \), \( \cos \), and \( \tan \) are vital in calculus, often appearing in differential equations due to their repetitive cyclical behavior.

The function \( \sin(\pi s) \) indicates a sinusoidal or wave-like pattern. In solving the differential equation \( \frac{dy}{ds} = \sin(\pi s) \), recognizing these patterns helps simplify the problem.

• The derivative of \( \sin(x) \) is \( \cos(x) \), and vice versa, flipping roles between sine and cosine when differentiated and integrated.
• Adjusting the argument with constants like \( \pi \) requires accounting for such changes during integration or differentiation.

Trigonometric functions model many real-world phenomena like sound waves, light waves, and tides, making them essential for scientific applications. In mathematics, their properties are leveraged in various equations to describe cyclical processes and patterns.

General Solution of Differential Equations

The general solution of a differential equation represents a family of functions rather than a single function. This is because the process of integration introduces an arbitrary constant, known as the constant of integration, \( C \).

In the solution of \( \frac{d y}{d s} = \sin(\pi s) \), we integrated to find:
\[ y(s) = -\frac{1}{\pi}\cos(\pi s) + C \]

• The term \( -\frac{1}{\pi}\cos(\pi s) \) provides the specific structure tied to the sine function in the differential equation.
• Adding \( C \) ensures we capture every possible vertical shift of the function, accounting for countless solutions.

This general solution highlights flexibility, as different initial conditions or constraints can specify particular solutions from this family by determining \( C \).
The concept of a general solution is crucial to fully capturing the implications of differential equations, especially in modeling situations where exact starting points are not fixed.