Suppose two students are memorizing the elements on a list according to the differential equation dL dt = 0.5(1 − L) where L represents the fraction of the list that is memorized at any time t given in hours. (a) What are the units in this scenario for dL dt ? (b) If one of the students knows one-third of the list at time t = 0, and the other student knows none of the list, which student is learning most rapidly at this instant? Explain your reasoning mathematically. (c) What kind of trend does this differential equation predict for someone who begins with the list completely memorized? Explain.
For the differential equation dy dt = 1 − y2 (a) Sketch a slope field for this differential equation by hand. Include t values [−2, 2] and y values [−2, 2] (your axes should be a 4×4 grid). (b) Describe any shortcuts or patterns you used to make the task easier. (c) Sketch two different possible solutions for y(t) on the axes you drew in part (a).
For each part, write a differential equation that satisfies the given criteria (if possible). If such a differential equation can't exist, explain why. (a) An ordinary, linear differential equation of order 3. (b) A ordinary, nonlinear differential equation of order 2 . (c) An ordinary differential equation for which y = t2 is a solution.
A slope field for the differential equation dy dt = 0.5(y + t) is shown below. (a) For the initial condition of y(0) = 1, sketch on the slope field what you think the solution function will look like. (b) Estimate, using your sketch, the value of y(2). (c) Use Euler's method with a step size of 0.5 to predict the value of y(2) with the initial condition y(0) = 1.
Consider the differential equation dy dt = −t y (a) Using the Separation of Variables method, find the general solution for this differential equation. (b) Given the initial condition y(0) = 4, find the particular solution for this differential equation. (c) With this particular solution, determine the exact value of y(1).
For each of the following differential equations, determine whether Separation of Variables could be used to find the general solution and explain why or why not. Do NOT actually find the solution. (a) dy dt = 2y − 3e−t (b) dy dt = −0.2(75 − y) (c) dy dt = y2 + 1 (d) dy dt = ety − cos(t)