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?[I] \(y^{\prime}=y(x)\)

Short answer

The critical point is at \( C = 0 \), where the solution \( y(x) = 0 \) is stable.

Step-by-step solution

Understand the Differential Equation

The given differential equation is \( y' = y(x) \), which is a simple first-order linear differential equation. This means that the derivative of \( y \) with respect to \( x \) is equal to \( y \).

Solve the Differential Equation

To find the family of solutions, we solve the equation \( y' = y(x) \). The solution to this is \( y(x) = Ce^{x} \), where \( C \) is the constant of integration. This constant will change depending on the initial condition for each solution curve.

Apply the Initial Conditions

Apply the range of initial conditions \( y(t=0) = -10 \) to \( y(t=0) = 10 \), increasing by 2, to determine the different values of \( C \). For each initial condition, solve for \( C \) using \( C = y(0)/e^0 = y(0) \).

Graph the Family of Solutions

Graph each solution \( y(x) = Ce^{x} \) for \( C = -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10 \). Use a graphing calculator or software to visualize the family of exponential curves. Since all solutions are ?cexponential, they all increase (or decrease) exponentially based on the sign of \( C \).

Identify the Critical Point

Observe the graph for any changes in the behavior of the solutions. The critical point is where the behavior of the solution changes from decreasing to increasing, which occurs at \( C = 0 \). At this point, the solution \( y(x) = 0 \) is a horizontal line, representing a steady state or equilibrium.

Key concepts

Initial Conditions

Initial conditions are essential when dealing with differential equations. They are specific values provided for a function and its derivatives at a particular point, used to find a unique solution within a family of solutions. In the context of the problem, initial conditions are specified as values of \( y(t = 0) \) ranging from -10 to 10, with increments of 2.

By applying these initial conditions to the general solution \( y(x) = Ce^{x} \), we can determine distinct constants \( C \) for each scenario. This essentially personalizes the general solution to form specific solutions for each condition. Without initial conditions, we can only describe the general behavior of a solution, not its exact form.

Family of Solutions

A family of solutions to a differential equation is a set of functions that are solutions to the equation under different conditions. The solution \( y(x) = Ce^{x} \) results in a family of solutions because the constant \( C \) varies.

Each value of \( C \) corresponds to a particular solution that fits the differential equation. This variability allows the solution to describe a wide range of behaviors and states. Here, each initial condition corresponds to a different value of \( C \) and thus a different solution in the family. Understanding how \( C \) impacts behavior is crucial because each alteration represents a unique pathway of possible outcomes.

Graphing Solutions

Graphing is a powerful visual tool that helps interpret the solutions to differential equations. By plotting \( y(x) = Ce^{x} \) for each value of \( C \) determined by the initial conditions, you construct the family of solution curves.

Each curve in the graph provides insight into how different conditions affect the growth or decay of the function. The exponential nature is evident as some curves rise steeply while others fall.

• When \( C > 0 \), the graph depicts exponential growth.
• When \( C < 0 \), it shows exponential decay.

Graphing helps in identifying trends and understanding dynamic changes across the family of solutions.

Equilibrium Point

The equilibrium point in a differential equation is where the values become constant, and there is no change over time. In this context, at \( C = 0 \), the solution becomes \( y(x) = 0 \), which offers an equilibrium point.

At this critical point, the solution does not increase or decrease. Instead, it maintains a steady state, depicted by a horizontal line on the graph. This is significant because it signifies stability within the system modeled by the equation. Finding equilibrium points is crucial for understanding the long-term behavior of systems and identifying states of balance.