Consider the population model dP/dt = 0.3P(1 − P/12.5) If P(0) < 12.5, will the population ever reach 13? Explain how you came to your conclusion.
For this problem, use the coffee cooling equation given below, where C represents the temperature of the coffee in degrees Fahrenheit, and t is time in minutes. dC dt = −0.4C + 28 (a) How long will take a cup of hot coffee that is initially 180 degrees to cool down to 100 degrees? (Hint: You can solve the ODE for the particular solution C(t) ) (b) Does this rate of change equation also make sense for predicting the future temperature for a glass of iced tea? Why or why not? (Hint: try plotting a slope field for this differential equation. What happens if the initial temperature is cool?)
A particular disease can spread in one of various ways, three of which are shown below. Let S(t) represent the number of individuals infected at time t. Let S0 represent the initial number of people infected (that is, S(0) = S0), and suppose the total population is 200. Let k be some constant. Three models representing that rate at which a population is infected are given below: Model 1: dS dt = k Model 2: dS dt = kS Model 3: dS dt = k(200 − S) (a) For each model, write a 1-2 sentence explanation using the problem context describing the rate at which the population is infected. (b) Match each graph below with the model it represents.
Find the general solution to each of the following differential equations using separation of variables or integrating factors. Give a reason as to why you used the method you chose over the other. 2 points each. (a) dy dt = 2y − t (b) dP dt = 2P t (c) dy dx = ysin(x)
Identify which of the following differential equations are exact. 1 point each. 3yt2 + (y3 + t3)dy dt = 0 ln(y)dy dt = 0 tsin(y) + t2cos(y)2 dy dt = 4t dy dt = t2 y3 t y + y t dy dt = 0 t2 + y3 + (y2 + t3)dy dt = 0
Using separation of variables, solve the differential equation, e−ysin(x) − dy dxcos2(x) = 0 Use C to represent the arbitrary constant. y =
Using separation of variables, solve the differential equation, (1 + x6)dy dx = x5 y Use C to represent the arbitrary constant. y2 =
Find an equation of the curve that satisfies dydx = 18yx5 and whose y-intercept is 3. y(x) =
Consider the first order differential equation y′ + t t2−9 y = et t−6 For each of the initial conditions below, determine the largest interval a < t < b on which the existence and uniqueness theorem for first order linear differential equations guarantees the existence of a unique solution. a. y(−7) = −0.5. help (inequalities) b. y(−2.5) = −0.5. help (inequalities) c. y(0) = 0. help (inequalities) d. y(3.5) = −0.5. help (inequalities) e. y(11) = 2.6. help (inequalities)
For the differential equation dy dx = y2−1 does the existence/uniqueness theorem guarantee that there is a solution to this equation through the point (−2, 1)? (2, −1)? (0, 7)? (−4, 3)?
Solve 4xy′ − 8y = x−2, y(1) = −9. (a) Identify the integrating factor, α(x). α(x) = (b) Find the general solution. y(x) = Note: Use C for the arbitrary constant. (c) Solve the initial value problem y(1) = −9. y(x) =