13. Intro to Differential Equations

Explain how the growth rate function determines the solution of a population model.

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁eᵗ + C₂teᵗ; u''(t) - 2u'(t) + u(t) = 0

Solve the following system of differential equations by systematic elimination:

$$\frac{\partial x}{\partial t}=2x-y$$$$\frac{\partial y}{\partial t}=x$$
Which of the following gives the general solution for $x\left(t\right)$?

State the order of the differential equation and indicate if it is linear or nonlinear. 

$\frac{d^{2}y}{d{x}^2}-\left(\frac{dy}{dx}\right)(1-x)=2$

The general solution of a first-order linear differential equation is y(t) = Ce⁻¹⁰ᵗ − 13. What solution satisfies the initial condition y(0) = 4?

Which of the following is the general solution to the differential equation $\frac{dy}{dx}=9x^2{y}^2$?

22–25. Equilibrium solutions Find the equilibrium solutions of the following equations and determine whether each solution is stable or unstable.

y′(t) = y(3+y)(y-5)

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.

y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt

Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.

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

y''(t) = teᵗ, y(0) = 0, y'(0) = 1

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

p'(x) = 2/(x² + x), p(1) = 0

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

(Use of Tech) Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y'(t) = -kyⁿ(t), where y(t) is the concentration of the compound, for t ≥ 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume the initial concentration of the compound is y(0) = y₀ > 0.

c. Let y₀ = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions. 

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.

Consider the following differential equation: $x\frac{dy}{dx}-y=x^2\sin \left(x\right)$. Which of the following is the correct integrating factor to solve this equation?