Domanda del libro di testo
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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Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 9, Problema 9.1.39
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
Guida verificata passo dopo passo1
Identify the differential equation and initial condition: \(y' = 2x \exp(x^2)\) with \(y(0) = 2\).
Set up Euler's method formula: \(y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x\), where \(f(x, y) = 2x \exp(x^2)\) and \(\Delta x = 0.1\).
Calculate the number of steps needed to reach \(x^* = 1\) starting from \(x_0 = 0\) using \(\text{steps} = \frac{x^* - x_0}{\Delta x} = \frac{1 - 0}{0.1} = 10\) steps.
Iteratively apply Euler's method: for each step \(n\) from 0 to 9, compute \(y_{n+1} = y_n + 2x_n \exp(x_n^2) \cdot 0.1\) and update \(x_{n+1} = x_n + 0.1\).
To find the exact solution at \(x = 1\), integrate the differential equation: \(y = \int 2x \exp(x^2) \, dx + C\). Use substitution \(u = x^2\) to solve the integral, then apply the initial condition \(y(0) = 2\) to find \(C\).
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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 next value by moving in small steps (dx). This iterative process continues until reaching the desired point.
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Euler's Method
Initial Value Problem (IVP)
An initial value problem specifies a differential equation along with a starting point (initial condition) for the solution. The solution curve is uniquely determined by this initial value, allowing numerical methods like Euler's to approximate the solution over an interval.
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Percorso guidato
05:03Initial Value Problems
Exact Solution of Differential Equations
The exact solution is an explicit formula that satisfies the differential equation and initial condition. Finding it often involves integration or known solution techniques. Comparing the exact solution to numerical approximations helps assess the accuracy of methods like Euler's.
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04:00Solutions to Basic Differential Equations
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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*.
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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
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Domanda del libro di testo
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)?
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Domanda del libro di testo
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
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