Given the initial value problem $\frac{dy}{dx} = x + y$ with $y(0) = 1$. Using Euler's forward method with a step size $h = 0.1$, the value of $y(0.2)$ is ________ (round off to 2 decimal places).
Answer Key:
1.22
Step-by-Step Solution & Explanation
Euler's forward formula is $y_{n+1} = y_n + h f(x_n, y_n)$ with $f(x, y) = x + y$.
Given $x_0 = 0, y_0 = 1, h = 0.1$:
1. First step ($x_1 = 0.1$):
$y_1 = y_0 + h (x_0 + y_0) = 1 + 0.1(0 + 1) = 1 + 0.1 = 1.1$
2. Second step ($x_2 = 0.2$):
$y_2 = y_1 + h (x_1 + y_1) = 1.1 + 0.1(0.1 + 1.1) = 1.1 + 0.1(1.2) = 1.1 + 0.12 = 1.22$
Thus, $y(0.2) = 1.22$.