Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Sketch a representative family of solutions for each of the following differential equations. (a) \(\frac{d y}{d t}=\sin t\) (b) \(\frac{d y}{d t}=\sin y\)

Step-by-Step Solution

Answer

This question is about finding the family of solutions for two differential equations and sketching them. In each case, we integrate to find the family of solutions. Specifically, the solutions for the differential equations \(\frac{d y}{d t}=\sin t\) is \(-\cos t + C\) for part (a) and for \(\frac{d y}{d t}=\sin y\) cannot be directly integrated, but is roughly a sequence of S-shaped curves skewed towards the t-axis for part (b).

Step 1 - Solve the differential equations

(a) The function to integrate is \(\sin t\). By the basic principle of calculus, \(\int \sin t \, dt = -\cos t + C\), where \(C\) is the constant of integration. This gives the general solution for part (a).\n (b) For part (b), the differential equation cannot be solved directly as in part (a). However, solutions for \(\frac{d y}{d t}=\sin y\) can approximate \(\sin y \) being \(y\) or \(-y\), so would be represented by a series of S-shaped curves skewed in the direction of increasing \(t\). This provides an approximate sketch for the family of solutions.

Step 2 - Sketch the solutions

(a) The function \(-\cos t + C\) on an xy-plane gives a wave which peaks at \(C+1\) and troughs at \(C-1\). As \(C\) varies, the wave moves up and down the y-axis, representing a family of solutions. In particular, for \(C=0\), the sketch is simply the curve of function \(-\cos t\).\n (b) The function \(\sin y\) also gives a wave-like graph, but with its shape skewed in the direction of increasing \(t\). Here, the behavior of the graph changes according to \(y\) as it oscillates between \(-1\) and \(1\). Each different curve, shifted up or down the t-axis, is a different family of solutions.

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They serve as a cornerstone of modeling in the natural sciences and engineering because they can describe the rate at which certain variables change over time or space. For instance, the exercise provided requires sketched solutions for two such equations:
For \(\frac{d y}{d t}=\sin t\), we're dealing with a standard first-order differential equation, where the rate of change of the function y(t) at any time t is given by the sine of t. Integrating this equation, we obtain a sinusoidal function shifted vertically by a constant value, from which the family of solutions is generated.
The second problem, \(\frac{d y}{d t}=\sin y\), is a bit trickier as it doesn't have a straightforward integration like the former. This non-linear differential equation suggests the rate of change of y with respect to t is governed by the sine of y itself, leading to a distinct set of solutions manifesting as S-shaped curves. Each curve in the respective family has its own particular character depending on the initial conditions set by the constant of integration.

Integral Calculus

Integral calculus is the process of finding a function when its derivative is known, which is the inverse operation of differentiation. It is fundamentally used in finding the area under curves and solving differential equations.

Application in Differential Equations

When given a differential equation like \(\frac{d y}{d t}=\sin t\), one of our main tools for finding a solution is to integrate both sides, which gives us \(y(t) = -\cos t + C\). Here, \(C\) represents the constant of integration, which corresponds to the infinite number of solutions—a 'family of solutions'—as \(C\) can take any value. In our exercise, this integration process helps in sketching the graph of the function that solves the differential equation.

Family of Solutions

The 'family of solutions' in the context of differential equations refers to a set of functions that all satisfy the given differential equation. Each member of this family is defined by the constant of integration (\(C\) in our case), which can take on an infinite range of values.

Variations in the Family

In the exercise, for part (a), changing the value of \(C\) essentially slides the function \(y(t) = -\cos t + C\) up or down on the graph, since each value of \(C\) changes the initial position of the sinusoidal function. This visual representation is crucial because it helps to understand the variety of solutions that can arise from a single differential equation depending on initial conditions or boundaries set by real-world scenarios.

Sketching Graphs of Functions

Sketching graphs is an essential method for visualizing the behaviour of functions and their respective family of solutions. This visual approach helps to comprehend the abstract mathematical concepts by seeing them in a more tangible form.

Sketching from Differential Equations

For \(\frac{d y}{d t}=\sin t\), upon integration, we visualize the graph as a series of waves. Each member of the family of solutions is represented by the same wave pattern shifted vertically. For \(\frac{d y}{d t}=\sin y\), the exercise suggests a more complex pattern of S-shaped curves, with the 'wave' oriented along a different axis. By meticulously sketching these patterns, taking into account the effect of the constant of integration, one can gain a deeper conceptual understanding of the differential equation's solutions and their potential application.