Final answer:
To match differential equations with their corresponding phase lines, analyze the qualitative behavior indicated by the phase line, find a differential equation whose solutions exhibit similar features, identify knowns and unknowns, solve the equation, and check if the solution is reasonable.
Step-by-step explanation:
To determine which differential equation corresponds to each phase line, one must analyze the behavior of the solutions as indicated by the phase line. This analysis typically involves identifying stable and unstable equilibria, and the direction of change between these points. By doing so, we can match the graph's qualitative behavior with the types of solutions we expect from specific differential equations. For example, let's consider a phase line with two stable equilibria separated by an unstable one. The corresponding differential equation would likely be a non-linear second-order differential equation that allows for multiple equilibria, with the middle point being an inflection point where stability changes.
Key steps to match phase lines with differential equations include:
Identifying knowns and unknowns from the problem
Solving the differential equation to find the unknowns
Checking if the solution is reasonable in terms of behavior, units, and values
To confirm your choices are correct, align the characteristics of the phase line with the expected qualitative behavior derived from solving the differential equation. If the phase line indicates that populations grow towards a certain equilibrium, the corresponding differential equation should be one where the solution approaches this equilibrium as time tends to infinity.