1. **Problem Statement:** Solve the system of differential equations using Euler's method with initial conditions $y_1(0) = 4$, $y_2(0) = 6$, step size $h = 0.5$, and integrate up to $x = 2$. 2. **Euler's Method Formula:** For a system $\frac{dy}{dx} = f(x,y)$, the Euler update is: $$y_{n+1} = y_n + h f(x_n, y_n)$$ where $h$ is the step size. 3. **Important Rules:** - Use the initial values to compute derivatives. - Update each $y_i$ using its derivative. - Repeat until reaching $x=2$. 4. **Assuming the system is:** $$\frac{dy_1}{dx} = f_1(x,y_1,y_2), \quad \frac{dy_2}{dx} = f_2(x,y_1,y_2)$$ Since the system functions $f_1$ and $f_2$ are not provided, we cannot compute numerical values. 5. **General Euler steps:** - At $x_0=0$, $y_1=4$, $y_2=6$ - Compute $f_1(x_0,y_1,y_2)$ and $f_2(x_0,y_1,y_2)$ - Update: $$y_1^{(1)} = y_1^{(0)} + h f_1(x_0,y_1^{(0)},y_2^{(0)})$$ $$y_2^{(1)} = y_2^{(0)} + h f_2(x_0,y_1^{(0)},y_2^{(0)})$$ - Repeat for $x_1=0.5$, $x_2=1.0$, $x_3=1.5$, $x_4=2.0$ **Final answer:** Without explicit differential equations, the numerical solution cannot be completed. Please provide the system $\frac{dy_1}{dx}$ and $\frac{dy_2}{dx}$ functions to proceed.