A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ
a. Show that the left side of the equation can be written as the derivative of a single term.

Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.

e. Discuss some possible shortcomings of this model. Why might the carrying capacity be either greater than or less than the value predicted by the model?

Logistic growth parameters A cell culture has a population of 20 when a nutrient solution is added at t=0. After 20 hours, the cell population is 80 and the carrying capacity of the culture is estimated to be 1600 cells.

c. After how many hours does the population reach half of the carrying capacity

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.

c. When does the temperature of the coffee reach 50°C?

A predator-prey model Consider the predator-prey model

x′(t) = −4x + 2xy, y′(t) = 5y − xy

c. Find the equilibrium points for the system.

A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ

c. Find the solution that satisfies the condition y(1) = 0

33–42. Solving initial value problems Solve the following initial value problems.

y'(x) = 4 sec² 2x, y(0) = 8

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

y'(t) = t lnt + 1

Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.

b. Use the solution of the logistic equation and the 2050 projected population to determine the carrying capacity.