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 9.1.22

Chapter 9, Problem 9.1.22

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)?

Verified step by step guidance

Identify the differential equation and initial condition: \(\frac{dy}{dx} = \frac{y}{x}\) with \(y(1) = 2\).

Set the step size \(\Delta x = 0.2\) and note that you want to estimate \(y(2)\) starting from \(x=1\).

Apply Euler's method iteratively: for each step, calculate the slope \(f(x_n, y_n) = \frac{y_n}{x_n}\), then update \(y\) using \(y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x\) and increment \(x\) by \(\Delta x\).

Perform the iterations from \(x=1\) to \(x=2\) in steps of \(0.2\), updating \(y\) at each step using the formula above.

To find the exact value of \(y(2)\), solve the differential equation analytically by separating variables: \(\frac{dy}{y} = \frac{dx}{x}\), integrate both sides, apply the initial condition to find the constant, and then evaluate the solution at \(x=2\).

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Key Concepts

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

Euler's Method

Euler's method is a numerical technique to approximate solutions of first-order differential equations. It uses a step size (dx) to incrementally estimate the value of the function by applying the slope given by the differential equation at each step. This method is especially useful when an exact solution is difficult to find.

Solving First-Order Differential Equations

A first-order differential equation relates a function and its derivative. To solve it exactly, one often separates variables or uses an integrating factor. For the equation y' = y/x, separating variables allows integration to find the explicit formula for y(x), which can then be evaluated at any point.

Initial Value Problems (IVP)

An initial value problem specifies the value of the solution at a particular point, such as y(1) = 2. This condition is essential for determining the unique solution to a differential equation. Both numerical methods like Euler's and exact solutions rely on this initial condition to find or approximate y at other points.

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Initial Value Problems

Related Practice

Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

Textbook Question

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

Textbook Question

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

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Textbook Question

Sailing A sailboat is running along a straight course with the wind providing a constant forward force of 50 lb. The only other force acting on the boat is resistance as the boat moves through the water. The resisting force is numerically equal to five times the boat’s speed, and the initial velocity is 1 ft/sec. What is the maximum velocity in feet per second of the boat under this wind?

Textbook Question

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

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Textbook Question

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' = 1 + y², y(0) = 0, dx = 0.1, x* = 1

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