Question

Using Euler’s MethodIn 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

Accepted Answer

Identify the differential equation and initial condition: $$y' = x(1 - y)$$ with $$y(1) = 0. $$Here, $$y'$$ represents the derivative $$\frac{dy}{dx}$$, and the initial point is at $$x = 1$$ with $$y = 0. $$Set the step size $$\Delta x = 0.2. $$Euler's method uses the formula $$y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n)$$, where $$f(x, y) = y' is $$the right-hand side of the differential equation. Calculate the first approximation: Evaluate $$f(x_0$, y_0$) = x_0$ (1 - y_0$) at $$the initial point $$(x_0$, y_0$) = (1, 0)$$, then compute $$y_1$$ = $$y_0 + 0.2$$ 	imes f($$x_0$$, $$y_0$$)$$. Calculate the second approximation: Update x_1$ = x_0$ + 0.2$$, then evaluate $$f(x_1$, y_1$)$$ using the previously found $$y_1$$, and compute $$y_2$$ = $$y_1 + 0.2$$ 	imes f($$x_1$$, $$y_1$$)$$. Calculate the third approximation: Update x_2$ = x_1$ + 0.2$$, evaluate $$f(x_2$, y_2$)$$, and compute $$y_3$$ = $$y_2 + 0.2$$ 	imes f($$x_2$$, $$y_2$$)$$. After these steps, find the exact solution by solving the differential equation analytically and compare the approximations to assess accuracy.

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Key Concepts

Euler's Method

Initial Value Problem (IVP)

Exact Solution and Error Analysis