Ch. 9 - First-Order Differential Equations

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 9 - First-Order Differential Equations

Problem 9.2.20

Chapter 9, Problem 9.2.20

Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.

dy/dx + xy = x, y(0) = -6

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Identify the type of differential equation. The given equation is \( \frac{dy}{dx} + xy = x \), which is a first-order linear differential equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \), where \( P(x) = x \) and \( Q(x) = x \).

Find the integrating factor (IF) using the formula \( \mu(x) = e^{\int P(x) \, dx} \). Here, calculate \( \mu(x) = e^{\int x \, dx} \).

Multiply both sides of the differential equation by the integrating factor \( \mu(x) \) to rewrite the left side as the derivative of the product \( \mu(x) y \). This gives \( \frac{d}{dx} [\mu(x) y] = \mu(x) Q(x) \).

Integrate both sides with respect to \( x \) to find \( \mu(x) y = \int \mu(x) Q(x) \, dx + C \), where \( C \) is the constant of integration.

Use the initial condition \( y(0) = -6 \) to solve for \( C \), then solve for \( y \) explicitly by dividing both sides by \( \mu(x) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Linear Differential Equations

These are differential equations of the form dy/dx + P(x)y = Q(x). They can be solved using an integrating factor, which simplifies the equation into an exact derivative, allowing integration to find the general solution.

Integrating Factor Method

The integrating factor is a function, usually denoted μ(x) = e^(∫P(x)dx), used to multiply both sides of a linear differential equation. This transforms the left side into the derivative of (μ(x)y), making the equation easier to integrate and solve.

Initial Value Problems (IVP)

An IVP specifies the value of the solution at a particular point, such as y(0) = -6. After finding the general solution, the initial condition is used to determine the specific constant, yielding a unique solution that satisfies both the differential equation and the initial condition.

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Initial Value Problems

Related Practice

Textbook Question

Solve the Bernoulli equations in Exercises 29–32.

y' - y = xy²

Textbook Question

What integral equation is equivalent to the initial value problem y' = f(x), y(x₀) = y₀?

Textbook Question

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

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Textbook Question

Using Euler’s Method

y' = y²(1+2x), (y-1) = 1, dx = 0.5

Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

e²ˣy' + 2e²ˣ y = 2x

Textbook Question

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

a. (1/cosx) ∫ cos x dx = tan x + C