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

Integral Equations

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

and initial condition for y.

y = 1 + ∫₀ ͯ y(t) dt

In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.

y′ = √x/y, y > 0, y(0) = 1, dx = 0.1, x* = 1

In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.

y' = 2xexp(x²) , y(0) = 2, dx = 0.1, x* = 1

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

t dy/dt + 2y = t³, t > 0, y(2) = 1

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = x(1-y), y(1) = 0, dx = 0.2

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

y' + (tanx)y = cos²x, -π/2 < x < π/2

Use Euler’s method with dx = 0.2 to estimate y(1) if y′ = y and y(0) = 1. What is the exact value of y(1)?

Carbon monoxide pollution An executive conference room of a corporation contains 4500 ft³ of air initially free of carbon monoxide. Starting at time t = 0, cigarette smoke containing 4% carbon monoxide is blown into the room at the rate of 0.3 ft³/min. A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0.3 ft³/min. Find the time when the concentration of carbon monoxide in the room reaches 0.01%.

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

In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.

y' = 2y²(x-1), y(2) = -1/2, dx = 0.1, x* = 3

Show that the solution of the initial value problem

y' = x + y, y(x₀) = y₀

is

y = -1 -x + (1 + x₀ + y₀) exp(x-x₀).

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

xdy/dx + y = e ͯ, x > 0

Use Euler’s method with dx = 1/3 to estimate y(2) if y′ = x sin y and y(0) = 1. What is the exact value of y(2)?

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = 2y/x, y(1) = -1, dx = 0.5

Integral Equations

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

and initial condition for y.

y = ∫₁ ͯ 1/t dt

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

θ dy/dθ + y = sin θ, θ > 0, y(π/2) = 1

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

(1+x)y' + y = √x

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = y²(1+2x), (y-1) = 1, dx = 0.5

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = 2xy + 2y, y(0) = 3, dx = 0.2

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

(x+1) dy/dx - 2 (x² + x)y = exp(x²) / (x+1), x > -1, y(0) = 5

Use Euler’s method with dx = 0.2 to estimate y(2) if y′ = y/x and y(1) = 2. What is the exact value of y(2)?

Solve the Bernoulli equations in Exercises 29–32.

x²y' + 2xy = y³

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.

Is either of the following equations correct? Give reasons for your answers.

a. (1/cosx) ∫ cos x dx = tan x + C

b. (1/cosx) ∫ cos x dx = tan x + C / cos x

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

e²ˣy' + 2e²ˣ y = 2x

Solve the Bernoulli equations in Exercises 29–32.

y' - y = xy²

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

dy/dt + 2y = 3, y(0) = 1

Use Euler’s method with dx = 0.5 to estimate y(5) if y′ = y²/√x and y(1) = −1. What is the exact value of y(5)?