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Ch. 9 - First-Order Differential Equations
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 9 - First-Order Differential Equations문제 9.1.41
9장, 문제 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

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
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.
추천 영상:
05:03
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.
추천 영상:
04:00
Solutions to Basic Differential Equations
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