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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Identify the differential equation and initial condition: \(\frac{dy}{dx} = y\) with \(y(0) = 1\).

Set the step size \(\Delta x = 0.2\) and note that you want to estimate \(y(1)\), so you will perform steps from \(x=0\) to \(x=1\) in increments of 0.2.

Apply Euler's method formula iteratively: \(y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n)\), where \(f(x, y) = y\) in this case.

Calculate each successive \(y\) value starting from \(y_0 = 1\) at \(x_0 = 0\), then \(y_1\) at \(x_1 = 0.2\), \(y_2\) at \(x_2 = 0.4\), and so on until \(x_5 = 1\).

For the exact value of \(y(1)\), recognize that the solution to \(\frac{dy}{dx} = y\) with \(y(0) = 1\) is \(y = e^x\), so \(y(1) = e^1 = e\).

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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 (dx) to incrementally estimate the value of the function by moving along the slope given by the derivative. Starting from an initial condition, each new value is found by adding the product of the derivative and step size to the previous value.

Initial Value Problem (IVP)

An initial value problem specifies a differential equation along with a starting point (initial condition) for the function. Here, y' = y with y(0) = 1 means the solution curve passes through (0,1). The initial condition is essential for determining the unique solution to the differential equation.

Exact Solution of y' = y

The differential equation y' = y has the exact solution y = Ce^x, where C is a constant determined by the initial condition. Given y(0) = 1, the solution is y = e^x. Thus, the exact value of y(1) is e, approximately 2.718, which can be used to compare with the Euler's method estimate.