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' = 2xy + 2y, y(0) = 3, dx = 0.2

Accepted Answer

Identify the differential equation and initial condition: $$y' = 2xy + 2y$$ with $$y(0) = 3. $$Here, $$y'$$ represents the derivative of $$y$$ with respect to $$x. $$Set the step size $$\Delta x = 0.2$$ and initialize the first point as $$(x_0$, y_0$) = (0, 3). $$Use Euler's method formula to find the next value: $$y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x$$, where $$f(x, y) = 2xy + 2y is $$the right-hand side of the differential equation. Calculate the first three approximations by iterating the formula: 
- For $$n=0$$, compute $$y_1$$ using $$x_0$$ and $$y_0.
- $$For $$n=1$$, compute $$y_2$$ using $$x_1$$ = $$x_0$$ + \Delta x$$ and y_1$.
- For n=2$, compute y_3$ using x_2$ = x_1$ + \Delta x$$ and $$y_2. $$Find the exact solution by solving the differential equation analytically (e.g., using an integrating factor or separation of variables), then compare the exact values at $$x = 0.2, 0.4, 0.6$$ with your Euler approximations to assess accuracy.

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Concetti chiave

Euler's Method

Initial Value Problems (IVP)

Exact Solution and Error Analysis