13. Intro to Differential Equations

Blowup in finite time Consider the initial value problem y'(t) = yⁿ + 1, y(0) = y₀, where n is a positive integer.

b. Solve the initial value problem with n = 2 and y₀ = 1/√2.

2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.

y′(t) = √(y/t)

In Exercises 1–22, solve the differential equation.

y' = sin³ x cos² y

Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.

a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).

39–42. Special equations A special class of first-order linear equations have the form a(t)y'(t)+a'(t)y(t)=f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form

a(t)y'(t) + a'(t)y(t) = d/dt (a(t)y(t)) = f(t). 

Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems. 

(t² + 1)y′(t) + 2ty = 3t², y(2) = 8

In Exercises 125–128 solve the differential equation.

127. yy' = sec(y²)sec²(x)

A second-order equation Consider the equation

t² y′′(t) + 2ty′(t) − 12y(t) = 0

b. Assuming the general solution of the equation is

y(t) = C₁ tᵖ¹ + C₂ tᵖ²,

find the solution that satisfies the conditions

y(1) = 0, y′(1) = 7

A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time t = 0, an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of 500 mg/L. The inflow rate is 0.06 L/min. Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant.

a. Write an initial value problem that models the mass of the drug in the blood, for t ≥ 0.

23–26. Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t≥0 graph the solution, and determine the first month in which the loan balance is zero.

B′(t) = 0.004B − 800, B(0) = 40,000

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

(x²+y²)dx + xy dy = 0

Show that (0, 0) and (c/d, a/b) are equilibrium points. Explain the meaning of each of these points.

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.

c. Find the equilibrium points for the system.

x′(t) = −3x + 6xy, y′(t) = y − 4xy

11–16. Initial value problems Solve the following initial value problems.

y'(x) = −y + 2, y(0) = −2

17–18. {Use of Tech} Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function.

The population increases from 50 to 60 in the first month and eventually levels off at 150.

Solving Bernoulli equations Use the method outlined in Exercise 43 to solve the following Bernoulli equations.

c. y′(t) + y = √y