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Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 9 - First-Order Differential EquationsProblem 9.1.41
Chapter 9, Problem 9.1.41

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

Verified step by step guidance
1
Identify the differential equation and initial condition: \(y' = \frac{\sqrt{x}}{y}\) with \(y(0) = 1\).
Set the step size \(\Delta x = 0.1\) and the target point \(x^* = 1\). Determine the number of steps needed: \(n = \frac{x^* - 0}{\Delta x} = 10\) steps.
Apply Euler's method iteratively using the formula: \(y_{k+1} = y_k + \Delta x \cdot f(x_k, y_k)\), where \(f(x, y) = \frac{\sqrt{x}}{y}\).
Start with \(x_0 = 0\) and \(y_0 = 1\). For each step \(k\) from 0 to 9, compute \(y_{k+1}\) using the formula and update \(x_{k+1} = x_k + \Delta x\).
To find the exact solution at \(x^* = 1\), solve the differential equation analytically by separating variables and applying the initial condition, then evaluate the solution at \(x = 1\).

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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 to incrementally estimate the function's value by moving along the slope given by the differential equation. This method is especially useful when an exact solution is difficult to find.
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Euler's Method

Initial Value Problems (IVP)

An initial value problem specifies the value of the solution at a starting point, allowing the differential equation to be solved uniquely. Here, y(0) = 1 sets the initial condition, which is essential for applying Euler's method and finding the particular solution.
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Initial Value Problems

Exact Solution of Differential Equations

The exact solution is an explicit formula that satisfies the differential equation and initial condition. Finding it involves techniques like separation of variables or integration. Comparing the exact solution to the Euler approximation helps assess the accuracy of the numerical method.
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Solutions to Basic Differential Equations
Related Practice
Textbook Question

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)?

Textbook Question

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.


tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

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

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

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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