1. **Problem statement:** Solve the initial value problem (IVP) given by the differential equation $$y' = 10 \sin(x^2 + y^2)$$ with initial condition $$x(0) = 1$$ and step size $$h = 0.5$$ using the Euler method. 2. **Formula used:** The Euler method approximates the solution of an IVP by iterating the formula: $$y_{n+1} = y_n + h f(x_n, y_n)$$ where $$f(x,y) = y'$$ is the derivative function. 3. **Initial values:** - $$x_0 = 0$$ (since the initial condition is at $$x=0$$) - $$y_0 = 1$$ (given) - Step size $$h = 0.5$$ 4. **Calculate the first step:** - Compute $$f(x_0, y_0) = 10 \sin(0^2 + 1^2) = 10 \sin(1)$$ - Approximate $$\sin(1) \approx 0.84147$$ - So, $$f(x_0, y_0) \approx 10 \times 0.84147 = 8.4147$$ - Update $$y_1 = y_0 + h f(x_0, y_0) = 1 + 0.5 \times 8.4147 = 1 + 4.20735 = 5.20735$$ 5. **Calculate the second step:** - Update $$x_1 = x_0 + h = 0 + 0.5 = 0.5$$ - Compute $$f(x_1, y_1) = 10 \sin(0.5^2 + 5.20735^2) = 10 \sin(0.25 + 27.103) = 10 \sin(27.353)$$ - Since $$\sin$$ is periodic, reduce $$27.353$$ modulo $$2\pi$$: $$27.353 \mod 6.2832 \approx 27.353 - 4 \times 6.2832 = 27.353 - 25.1328 = 2.2202$$ - Approximate $$\sin(2.2202) \approx 0.795$$ - So, $$f(x_1, y_1) \approx 10 \times 0.795 = 7.95$$ - Update $$y_2 = y_1 + h f(x_1, y_1) = 5.20735 + 0.5 \times 7.95 = 5.20735 + 3.975 = 9.18235$$ 6. **Summary:** After two Euler steps with $$h=0.5$$ starting at $$y(0)=1$$, the approximate values are: - $$y(0.5) \approx 5.20735$$ - $$y(1.0) \approx 9.18235$$ **Final answer:** $$y(0.5) \approx 5.21$$ (rounded to two decimals)