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

step 1 In this video, we're going to be looking at the graphs of solutions to differential equations. S

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

Key Concepts

Existence and Uniqueness Theorem

This theorem guarantees that, under appropriate conditions (typically involving continuity and local Lipschitz continuity), a differential equation has a unique solution through a given initial point. For the equation dy/dt = e^t (y - 1)^2, if a candidate function’s graph passes through a point (t?, 1), then by the uniqueness theorem the solution must be unique and, in fact, if one solution is constant (like y(t) = 1), any other solution touching that value at t? would have to coincide with it. This property can rule out graphs that cross or touch y = 1 nontrivially while also deviating elsewhere, as they would violate uniqueness conditions.

Equilibrium Solutions

An equilibrium solution occurs when the derivative dy/dt equals zero regardless of t. In the given differential equation, setting y - 1 = 0 makes the right-hand side zero; hence y(t) = 1 is an equilibrium solution. If a proposed graph is non-constant yet passes through y = 1, then the behavior at that point must align with the zero-slope condition. A graph that deviates in slope or curvature at y = 1, or that only touches y = 1 momentarily, contradicts the necessary condition imposed by the differential equation and the corresponding equilibrium solution.

Consistency Between Graph Behavior and Differential Equation Requirements

A valid solution must have a tangent at every point that matches the value given by the differential equation. For dy/dt = e^t (y-1)^2, at points where y is near 1, the derivative is near zero. If a candidate graph does not exhibit this behavior—such as having a nonzero tangent when it should be horizontal, or having a discontinuity or cusp—it cannot be a solution. The inability of the graph's slope to conform to the prescribed behavior implied by the differential equation underscores its incompatibility.

Transcript

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