13. Intro to Differential Equations

Find the general solution to the differential equation.

$\frac{dy}{dx}=y\sqrt{x}$

A bad loan Consider a loan repayment plan described by the initial value problem

B'(t)=0.03B−600,B(0)=40,000,

where the amount borrowed is B(0)=\$40,000, the monthly payments are \$600, and B(t) is the unpaid balance in the loan.

b. What is the most that you can borrow under the terms of this loan without going further into debt each month?

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.

y(t) = sec² t/(2y), y(π/4) = 1

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.

a. Identify which equation corresponds to the predator and which corresponds to the prey.

x′(t) = −3x + xy, y′(t) = 2y − xy

In Exercises 1–22, solve the differential equation.

y' = (y²-1)x⁻¹

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.

y′(t) = -3y + 9, y(0) = 4

Explain how to solve a separable differential equation of the form

g(t)y'(y) = h(t)

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

a. Write an initial value problem for the mass of the substance.

A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.

An iron rod is removed from a blacksmith’s forge at a temperature of 900°C . Assume k=0.02 and the rod cools in a room with a temperature of 30°C When does the temperature of the rod reach 100°C? 

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.

y'(t) = yeᵗ, y(0) = −1

{Use of Tech} Chemical rate equations Let y(t) be t he concentration of a substance in a chemical reaction (typical units are moles/liter). The change in the concentration, under appropriate conditions, is modeled by the equation dy/dt=-ky^n for t≥0, where k>0 is a rate constant and the positive integer n is the order of the reaction.

b. Solve the initial value problem for a second-order reaction (n=2) assuming y(0)=y0.

45–48. General first-order linear equations Consider the general first-order linear equation y'(t)+a(t)y(t)=f(t). This equation can be solved, in principle, by defining the integrating factor p(t)=exp(∫a(t)dt). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes

p(t)(y′(t) + a(t)y(t)) = d/dt(p(t)y(t)) = p(t)f(t).

Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor.

y′(t) + (2t)/(t² + 1)y(t) = 1 + 3t², y(1) = 4

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

u'(x) = e²ˣ⁻ᵘ

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

a. y′(t) + y = 2y²

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.

b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.

x′(t) = 2x − xy, y′(t) = −y + xy