To solve the given differential equation using Euler’s numerical method with step size $h = 0.1$, we proceed as follows:

The given differential equation is: $\frac{dy}{dx} + 2xy = x, \quad y(0) = 2$

Rewriting it in standard form for Euler's method: $\frac{dy}{dx} = x - 2xy$

Define: $f(x, y) = x - 2xy$

Step-by-step process:

Initial conditions:

$x_0 = 0, \quad y_0 = 2$

We calculate $y$ iteratively at $x = 0.1, 0.2, \dots, 0.6$.

Iteration formula:

$y_{n+1} = y_n + h \cdot f(x_n, y_n)$

Here, $h = 0.1$.

Iterations:

1. At $x_0 = 0, y_0 = 2$: $f(0, 2) = 0 - 2(0)(2) = 0$ $y_1 = y_0 + h \cdot f(0, 2) = 2 + 0.1 \cdot 0 = 2$

2. At $x_1 = 0.1, y_1 = 2$: $f(0.1, 2) = 0.1 - 2(0.1)(2) = 0.1 - 0.4 = -0.3$ $y_2 = y_1 + h \cdot f(0.1, 2) = 2 + 0.1 \cdot (-0.3) = 2 - 0.03 = 1.97$

3. At $x_2 = 0.2, y_2 = 1.97$: $f(0.2, 1.97) = 0.2 - 2(0.2)(1.97) = 0.2 - 0.788 = -0.588$ $y_3 = y_2 + h \cdot f(0.2, 1.97) = 1.97 + 0.1 \cdot (-0.588) = 1.97 - 0.0588 = 1.9112$

4. At $x_3 = 0.3, y_3 = 1.9112$: $f(0.3, 1.9112) = 0.3 - 2(0.3)(1.9112) = 0.3 - 1.14672 = -0.84672$ $y_4 = y_3 + h \cdot f(0.3, 1.9112) = 1.9112 + 0.1 \cdot (-0.84672) = 1.9112 - 0.084672 = 1.826528$

5. At $x_4 = 0.4, y_4 = 1.826528$: $f(0.4, 1.826528) = 0.4 - 2(0.4)(1.826528) = 0.4 - 1.4612224 = -1.0612224$ $y_5 = y_4 + h \cdot f(0.4, 1.826528) = 1.826528 + 0.1 \cdot (-1.0612224) = 1.826528 - 0.10612224 = 1.72040576$

6. At $x_5 = 0.5, y_5 = 1.72040576$: $f(0.5, 1.72040576) = 0.5 - 2(0.5)(1.72040576) = 0.5 - 1.72040576 = -1.22040576$ $y_6 = y_5 + h \cdot f(0.5, 1.72040576) = 1.72040576 + 0.1 \cdot (-1.22040576) = 1.72040576 - 0.122040576 = 1.598365184$

7. At $x_6 = 0.6, y_6 = 1.598365184$: $f(0.6, 1.598365184) = 0.6 - 2(0.6)(1.598365184) = 0.6 - 1.9180382208 = -1.3180382208$ $y_7 = y_6 + h \cdot f(0.6, 1.598365184) = 1.598365184 + 0.1 \cdot (-1.3180382208) = 1.598365184 - 0.13180382208 = 1.46656136192$

Final Result:

$y(0.6) \approx 1.467 \, (\text{rounded to three decimal places})$

Would you like a detailed comparison with other methods or visual representation of the solution?

Related Questions:

1. How can this solution be checked using an exact or analytical solution?
2. What is the error associated with using Euler's method in this case?
3. How does the step size $h$ affect the accuracy of Euler's method?
4. Can higher-order methods like the Runge-Kutta method improve the accuracy here?
5. How does the behavior of the differential equation change for different initial conditions?

Tip: Euler's method is a simple numerical approach but can accumulate significant errors for large step sizes.