Using Euler's modified method to compute y for x=0.2, given that dy/dx=log(x+y) with initial condition y=1, x=0.

Steps to Solve

1. Set Up Initial Values Start with the initial condition, where $x_0 = 0$ and $y_0 = 1$. We want to find $y$ at $x = 0.2$ with a step size of $h = 0.1$.

2. Compute Intermediate Points Calculate the two steps required to reach from $x = 0$ to $x = 0.2$:

• First step: From $x = 0$ to $x = 0.1$
• Second step: From $x = 0.1$ to $x = 0.2$

3. First Step Calculation For the first step, calculate $f(x_0, y_0)$ where $f(x, y) = \log(x+y)$. Thus, $$ f(0, 1) = \log(0 + 1) = \log(1) = 0 $$

Now calculate: $$ y^* = y_0 + h f(x_0, y_0) = 1 + 0.1 \cdot 0 = 1 $$

Then calculate: $$ f(x^, y^) = f(0.1, 1) = \log(0.1 + 1) = \log(1.1) $$

Update $y_1$: $$ y_1 = y_0 + \frac{h}{2} \left( f(x_0, y_0) + f(x^, y^) \right) = 1 + 0.05 \left( 0 + \log(1.1) \right) $$

Calculating the value of $y_1$.

4. Second Step Calculation Repeat the first step calculation for the second interval: Compute $f(x_1, y_1)$: $$ f(0.1, y_1) = \log(0.1 + y_1) $$

Then calculate: $$ y^* = y_1 + h f(x_1, y_1) $$

Finally, update $y_2$: $$ y_2 = y_1 + \frac{h}{2} \left( f(x_1, y_1) + f(x^, y^) \right) $$

5. Final Result Now, after computing $y_2$, you will have the approximation of $y$ at $x = 0.2$ using Euler's modified method (Heun's method).