Math 100 Final - Differential Equations and Numerical Methods

Logistic growth, Qualitative Analysis, Eulers method

"What is a Differential Equation (DE)?"

"An equation relating a function $y(t)$ to its derivatives (e.g.

"How do you verify if a function is a solution to a DE?"

"Differentiate the function

"What is the differential equation for simple Exponential Growth?"

"$\frac{dy}{dt} = ky$"

"What is the solution to the equation $\frac{dy}{dt} = ky$?"

"$y(t) = y_0 e^{kt}$ (where $y_0$ is the initial value)."

"What does the parameter $K$ represent in the Logistic Equation $\frac{dy}{dt} = ry(1 - \frac{y}{K})$?"

"The Carrying Capacity (the maximum population the system can support)."

"How do you find the Equilibrium Solutions (Steady States) of a differential equation?"

"Set the derivative equal to zero ($\frac{dy}{dt} = 0$) and solve for $y$."

"What is Euler's Method used for?"

"To numerically approximate the solution of a differential equation when an exact algebraic solution is difficult or impossible."

"What is the iterative formula for Euler's Method?"

"$y_{n+1} = y_n + f(t_n

"In a Slope Field

what do the small line segments represent?"

"What happens to the accuracy of Euler's Method as you make the step size ($\Delta t$) smaller?"

"The accuracy generally increases (the error decreases)