Explain why the functions with the given graphs can't be solutions of the differential equation

step 1 In this question, you are asked to explain why given these two different graphs, why this differ

Explain why the functions with the given graphs can't be...

Key Concepts

Monotonicity and Slope Consistency

In the provided differential equation, the term (y-1)² is always nonnegative (and multiplied by the positive constant e²), ensuring that the derivative is nonnegative for all values of y. This implies that non-equilibrium solutions must be monotonically non-decreasing. If a graph shows behavior that decreases or has inconsistent slopes inconsistent with this monotonic increase, it cannot represent a solution to the differential equation.

Equilibrium (Stationary) Solutions

An equilibrium or stationary solution is one for which the derivative is zero, meaning that once the solution reaches this state it remains constant. In the given differential equation, when y equals 1 the derivative becomes zero. Therefore, any acceptable solution must either be identically 1 or approach it in a way that respects the stability and non-crossing properties of equilibrium solutions.

Existence and Uniqueness Theorem

This theorem asserts that, under appropriate conditions (such as the right-hand side of the differential equation being Lipschitz continuous), there exists exactly one solution passing through any given point. Consequently, two different solution curves cannot intersect, so any graph that appears to cross or merge with another violates this fundamental property of differential equations.

Autonomous Differential Equations

An autonomous differential equation is one in which the rate of change of the dependent variable depends solely on the dependent variable itself and not explicitly on the independent variable. This means that the slope at any point on a solution curve is determined only by the value of the dependent variable, ensuring uniform behavior of the solutions that depends exclusively on y.

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