Question

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from \(y(t=0)=-10\) to \(y(t=0)=10\) increasing by \(2 .\) Is there some critical point where the behavior of the solution begins to change?\([\mathrm{T}] y^{\prime}=t^{3}\)

Short answer

No critical point is observed; the parabolas shift vertically but retain the same shape.

Step-by-step solution

Understand the Differential Equation

The given differential equation is \( y' = t^3 \). This indicates that the rate of change of \( y \) with respect to \( t \) is \( t^3 \). Our task is to find a general solution form of \( y(t) \) and analyze its behavior given initial conditions.

Integrate the Differential Equation

To find the general solution, integrate both sides of the differential equation with respect to \( t \). The integration of \( y' \) gives \( y(t) = \int t^3 \, dt = \frac{t^4}{4} + C \), where \( C \) is the integration constant representing a family of solutions.

Apply Initial Conditions

We have multiple initial conditions from \( y(t=0) = -10 \) to \( y(t=0) = 10 \) increasing by 2. Substitute \( t = 0 \) and the respective \( y \)-value into \( y(0) = \frac{0^4}{4} + C \) to solve for \( C \). For example, if \( y(0) = -10 \), then \( -10 = 0 + C \) so \( C = -10 \). Repeat for each initial value: -10, -8, -6, ..., 10.

Graph the Family of Solutions

Using a graphing calculator or software, plot each equation \( y(t) = \frac{t^4}{4} + C \) for each \( C \) derived from the initial conditions. This will show a family of parabolic curves with varying vertical shifts.

Identify Critical Points

By observing the graphs generated in the previous step, identify where the behavior of the curves begins to change significantly. Look for changes in direction, asymptotes, or diverging lines. Since \( y(t) = \frac{t^4}{4} + C \) is a polynomial, no drastic behavior change is expected without further external factors, such as differentials leading to discontinuities.

Key concepts

Family of Solutions

A family of solutions in the context of differential equations refers to a set of functions that solve the equation and are distinguished by a constant parameter. Here, the differential equation is given by \( y' = t^3 \). When we integrate this equation with respect to \( t \), we obtain a general solution \( y(t) = \frac{t^4}{4} + C \). The constant \( C \) allows us to describe multiple solutions that differ graphically by vertical shifts. Each specific value of \( C \) represents a particular curve in the family of solutions. These curves are essentially translations of one another along the vertical axis by the constant \( C \). This concept is crucial for understanding how solutions to differential equations can vary across different initial conditions.

Initial Conditions

Initial conditions are specific values that provide a starting point to uniquely determine a particular solution from the family of solutions. In our exercise, the initial conditions are given as \( y(t=0)=-10, -8, -6, \ldots, 10 \). To find the specific solution corresponding to each initial condition, we substitute \( t=0 \) into the general solution \( y(t) = \frac{t^4}{4} + C \). Solving for \( C \) each time results in different specific equations, each represented by:

• \( C = -10 \)
• \( C = -8 \)
• \( C = -6 \)
• ... up to \( C = 10 \)

These specific solutions allow us to explore how a differential equation behaves under various starting conditions.

Rate of Change

The rate of change provided by the differential equation \( y' = t^3 \) describes how the function \( y(t) \) changes with respect to the variable \( t \). Here, the rate of change depends on \( t \) cubed, meaning it increases rapidly as \( t \) increases.

• At \( t=0 \), the rate of change is zero. This often implies a momentary pause in the variation of \( y \), assuming no other external influences.
• As \( t \) becomes positive or negative, \( y' \) grows quickly, and the curve of \( y(t) \) tends to steepen.

Understanding the rate of change is fundamental to predict how solutions develop over time, particularly in scientific contexts where differential equations model dynamic systems.

Graphing Differential Equations

When graphing differential equations, especially a family of solutions like \( y(t) = \frac{t^4}{4} + C \), the goal is to visualize the entire suite of solutions implied by different constants \( C \). This can be greatly aided by graphing calculators or software. By graphing:

• Each plot corresponds to a solution that arises from a specific initial condition.
• The graphs will generally appear as parabolic curves due to the \( t^4 \) term.

This visualization helps identify critical points, if any, where the nature of these parabolas might change due to behavior not initially considered in the function form. For this exercise, however, given the polynomial nature of the equation, significant changes are not expected without any additional forces or constraints applied to the system. Thus, understanding the graphical representation is crucial for spotting patterns and behaviors in these solutions.