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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 9.1.40
Chapter 9, Problem 9.1.40

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

Verified step by step guidance
1
Identify the differential equation and initial condition: \(y' = 2y^{2}(x-1)\) with \(y(2) = -\frac{1}{2}\).
Set the step size \(\Delta x = 0.1\) and the target point \(x^* = 3\). We will use Euler's method to approximate \(y(3)\) starting from \(x=2\).
Recall Euler's method formula: \(y_{n+1} = y_n + f(x_n, y_n) \Delta x\), where \(f(x,y) = 2y^{2}(x-1)\) is the derivative function.
Calculate successive values of \(y\) at each step by plugging in \(x_n\) and \(y_n\) into \(f(x,y)\), then update \(y_{n+1}\) using the formula until you reach \(x=3\).
To find the exact solution at \(x=3\), solve the differential equation analytically using separation of variables or an appropriate method, then substitute \(x=3\) into the exact solution.

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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. Starting from an initial condition, it uses the slope given by the differential equation to estimate the function's value at successive points by stepping forward in small increments (dx). This method is iterative and provides approximate values when exact solutions are difficult to find.
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Initial Value Problems (IVP)

An initial value problem specifies a differential equation along with a starting point (initial condition) that defines the solution uniquely. Here, y(2) = -1/2 sets the initial value at x = 2, which is essential for applying Euler's method and for determining the exact solution of the differential equation.
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Exact Solution of Differential Equations

The exact solution is an explicit function that satisfies the differential equation and initial condition without approximation. Finding the exact solution often involves techniques like separation of variables or integrating factors, allowing comparison with numerical approximations to assess their accuracy.
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Related Practice
Textbook Question

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