Problem 15 - 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) \(\frac{d y}{d t}=t^{2}-1\) (b) \(\frac{d y}{d t}=y^{2}-1\)

Step-by-Step Solution

Answer

Part (a): The differential equation was resolved via direct integration. The general solution is \(y = \frac{1}{3}t^{3} - t + C\) and constitutes a family of vertically shifted parabolas. Part (b): The non-separable differential equation's solutions were found by drawing a slope field. The representative lines in the slope field suggest that solutions are drawn towards equilibrium points \(y = -1\) and \(y = 1\).

Step 1: Determine the type of differential equations

Both the presented differential equations are first order, non-homogeneous equations as there is only a single derivative and a non-zero function on the right side of the equation.

Step 2: Solve part (a)

The differential equation \(\frac{d y}{d t}=t^{2}-1\) is non-separable. Use indefinite integration to find its general solution:

Step 3: General solution of part (a)

To integrate \(\frac{d y}{d t}=t^{2}-1\), recognize the right side as a polynomial in \(t\). The antiderivative of \(t^{2}\) is \(\frac{1}{3}t^{3}\) and the antiderivative of \(1\) is \(t\). Thus, the general solution is: \(y = \frac{1}{3}t^{3} - t + C\), where \(C\) denotes the arbitrary constant.

Step 4: Draw the family of solutions of part (a)

At this point, the student would want to use a graphing software or calculator to draw the family of solutions. These are parabolas with varying vertical shifts \(C\). For differing values of \(C\), interpret it as a vertical shift in the graph of \(y\) versus \(t\).

Step 5: Solve part (b)

The differential equation \(\frac{d y}{d t}=y^{2}-1\) is nonlinear and non-separable. In this case, one can draw a slope field to get a representational graph of the solutions.

Step 6: Draw the family of solutions of part (b)

Use a graphing tool to generate a slope field for the differential equation \(\frac{d y}{d t}=y^{2}-1\). The drawn representative lines are asymptotic around \(y = -1\) and \(y = 1\). Solutions which start at \(y = 1\) and \(y = -1\) remain constant. Solutions starting at other values of \(y\) evolve over time towards those equilibrium points.

Key Concepts

First Order Differential Equations

First order differential equations are equations that involve the first derivative of a function. In mathematical terms, they are expressed as \( \frac{dy}{dt} = f(t, y) \), where the equation contains the derivative \( \frac{dy}{dt} \). These equations are pivotal in describing how things change over time, such as velocity, growth rates, or decay. When tackling a first order differential equation, the goal is to find the function \( y(t) \) that satisfies the equation. This involves understanding the relationship between the variables and interpreting the given functions. First order equations can be categorized further into homogeneous or non-homogeneous equations, depending on whether a term with no derivative and no dependent variable appears in the equation.

Non-Homogeneous Equations

A non-homogeneous differential equation is one where the equation includes a term that is not dependent on the unknown function or its derivatives. In the exercises presented, both \( \frac{dy}{dt} = t^2 - 1 \) and \( \frac{dy}{dt} = y^2 - 1 \) are non-homogeneous. The presence of the constant \(-1\) indicates that the solutions will not pass through the origin unless specific initial conditions are met.These equations are typically more complex to solve than homogeneous equations because of their non-zero terms on the right-hand side. The general approach to solving non-homogeneous equations often involves finding a particular solution to the entire equation and then adding it to the general solution of the related homogeneous equation. While this might sound complicated, understanding the basics of integration and differentiation helps in solving these equations.

Indefinite Integration

Indefinite integration is a mathematical process used to find the antiderivative of a function. It's a key tool in solving first order differential equations, like \( \frac{dy}{dt} = t^2 - 1 \) from the exercise. The goal of indefinite integration is to reverse the process of differentiation, essentially finding the original function given its derivative. When you indefinitely integrate a function, you also include a constant \( C \), which represents an infinite set of solutions or a "family of solutions." This constant stems from the fact that differentiation removes any constant term.For instance, the indefinite integrals of the terms in \( t^2 \) and \(-1\) are \( \frac{1}{3}t^3 \) and \(-t \) respectively, leading to the general solution \( y = \frac{1}{3}t^3 - t + C \). This solution represents a family of curves, each differing by their vertical shift, \( C \).

Slope Field

A slope field is a visual representation of a differential equation, which helps to illustrate how solutions behave without explicitly solving the equation. For the equation \( \frac{d y}{d t} = y^2 - 1 \), drawing a slope field graph provides insight into its characteristics. In a slope field, tiny line segments are plotted across a plane to represent the slope given by the differential equation at various points \((t, y)\). This visual tool allows us to see patterns, like where solution curves might level out or become steeper.
In the case of the exercise, the slope field for \( \frac{d y}{d t} = y^2 - 1 \) shows lines that become horizontal at \( y = 1 \) and \( y = -1 \). These are equilibrium solutions, where the slope is zero, indicating that any solution curve reaching these points remains constant. This field helps to predict how other solutions will behave, showing trajectories leading towards or away from these lines.