Question

Explain why the functions with the given graphs cannot be solutions of the differential equation \(\frac{d y}{d t}=e^{t}(y-1)^{2}\)

Short answer

Question: Explain why the functions with the given graphs cannot be solutions of the differential equation \(\frac{d y}{d t} = e^{t}(y-1)^{2}\). Answer: The functions with the given graphs cannot be solutions of the differential equation if the graphs indicate that the function is decreasing with respect to \(t\) at any point. This is because the differential equation has a non-negative derivative, meaning the solutions must be non-decreasing. If a function's graph violates the non-decreasing property, it cannot be a solution to the differential equation.

Step-by-step solution

Analyze the given differential equation

We can start by looking at the given differential equation: \(\frac{d y}{d t}=e^{t}(y-1)^{2}\) One important aspect we can notice is that \((y-1)^2\) will always be non-negative since any square of a real number is non-negative. Also, \(e^t\) is an exponential function, always positive for any real value of \(t\). As a result, the product \(e^{t}(y-1)^{2}\) will always be non-negative.

Discuss the consequences of the properties in step 1 for the solutions of the differential equation

Since we established that \(\frac{d y}{d t}\) is non-negative for any values of \(t\) and \(y\), this means that the graph of any solution to the given differential equation must be non-decreasing. In other words, for any solution, the function value \(y(t)\) must either increase or stay constant with respect to \(t\). A function that violates this property cannot be a solution to the given differential equation.

Analyze the given graphs

To explain why the functions with the given graphs cannot be solutions to the differential equation, examine the given graphs carefully. Look for any instances where the graph indicates that the function is decreasing with respect to \(t\). If you find any such instances, this would imply that the function violates the non-decreasing property we derived in step 2. Consequently, the function cannot be a solution to the given differential equation. If the graphs do not provide enough information or are unclear, you may need to derive the functions represented by these graphs and check whether they satisfy the differential equation or not.

Conclusion

By examining the given graphs and checking if they satisfy the properties derived from the differential equation, we are able to explain why the functions with the given graphs cannot be solutions of the differential equation \(\frac{d y}{d t} = e^{t}(y-1)^{2}\). A function must satisfy the non-decreasing property and other conditions required by the differential equation in order to be a potential solution.

Key concepts

Exponential Functions

Exponential functions play a crucial role in understanding the behavior of solutions to differential equations. An exponential function typically has the form \( y = e^{kx} \) where \( e \) is the base of the natural logarithm, approximately equal to 2.71828, and \( k \) is a constant. These functions are known for their remarkable properties: they never take negative values, and the function \( e^x \) increases exponentially as \( x \) increases.

In solving differential equations, exponential functions often appear in the rate of change of a variable. For example, the rate of change of \( y \) with respect to \( t \) in the given equation—\( \frac{d y}{d t} = e^{t}(y-1)^{2} \)—suggests that the growth rate is not constant but changes in proportion to an exponential function of \( t \). This has significant implications for the behavior of solutions over time, as functions that grow too slowly or decrease over time simply cannot match the behavior dictated by an exponential growth rate. It is vital to recognize that, due to the nature of the function \( e^t \) being perpetually positive, solutions will have a certain characteristic shape on a graph: they will tend to grow faster as \( t \) increases, unless counterbalanced by other factors—in our case, the squared term \( (y-1)^2 \)—contained within the differential equation.

Non-negative Solutions

The concept of non-negative solutions is integral to the analysis of certain differential equations. A solution to a differential equation is a function that satisfies the equation for all values within a given domain. By 'non-negative' we mean that the function never takes on negative values; it is either positive or zero across its domain.

Considering the differential equation \( \frac{d y}{d t} = e^{t}(y-1)^{2} \), the term \( (y-1)^2 \) guarantees that, for any real value of \( y \), this part of the equation will yield a non-negative value, since squaring any real number cannot result in a negative value. When you combine this with the exponential term \( e^t \), which is also always positive, the product of the two terms will invariably be non-negative. Thus, the rate of change of \( y \) with respect to \( t \) is never negative, indicating that the function values of \( y(t) \) cannot decrease as \( t \) increases.

This property limits the types of functions that can be solutions to our differential equation. For example, functions that dip below the \( t \) axis or show decreasing trends are immediately excluded because they would require negative values of \( \frac{d y}{d t} \) at some points, which contradicts our established property of the differential equation.

Graph Analysis

Graph analysis is a visual tool that aids in understanding the functions that are solutions to differential equations. When presented with potential solutions in graphical form, analysis entails looking for key characteristics that match the requirements set by the differential equation. For the equation in question—\( \frac{d y}{d t} = e^{t}(y-1)^{2} \)—we expect to see a graph of a solution that is always non-decreasing.

By examining a graph, we search for sections where the function decreases; such sections would violate the non-negative rate of change established by the differential equation. Additionally, because of the presence of an exponential function, we would anticipate the rate of increase to not only be non-negative but potentially become more pronounced as \( t \) increases. Furthermore, intersection points with the line \( y = 1 \) are of particular interest, as at these points, the rate of change should be zero, leading to horizontal tangents on the graph.

Exercise Improvement Advice

In dealing with graph analysis, it is beneficial to annotate the graphs to highlight significant features explicitly. For instance, marking points where the graph intersects the line \( y = 1 \) and illustrating the slope behavior can clarify the reasons why a specific graph cannot represent a solution to the differential equation. An improved understanding of graph characteristics and their relationship to differential equations will enhance the students' ability to discriminate between possible solutions and non-solutions more effectively.