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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 9.1.41
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
Guida verificata passo dopo passo1
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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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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07:33
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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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 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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04:00Solutions to Basic Differential Equations
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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' = 2xexp(x²) , y(0) = 2, dx = 0.1, x* = 1
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y' = x(1-y), y(1) = 0, dx = 0.2
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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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Domanda del libro di testo
Use Euler’s method with dx = 1/3 to estimate y(2) if y′ = x sin y and y(0) = 1. What is the exact value of y(2)?
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