euler’s formula differential equations
[latex]
\[
\textbf{Euler’s Formula for Differential Equations:}
\]
\[
y_{n+1} = y_n + h f(x_n, y_n)
\]
\[
\text{where:}
\]
\begin{aligned}
&y_{n+1} \text{ is the next value of } y, \\
&y_n \text{ is the current value of } y, \\
&h \text{ is the step size, and} \\
&f(x_n, y_n) \text{ is the value of the derivative at } (x_n, y_n).
\end{aligned}
\[
\textbf{Example: Using Euler’s Formula for Differential Equations}
\]
\[
\text{Given the differential equation: } \frac{dy}{dx} = x + y, \quad y(0) = 1
\]
\[
\text{Calculate } y \text{ at } x = 0.1 \text{ using Euler’s method with step size } h = 0.1.
\]
\[
\textbf{Step 1: Initialize values.}
\]
\[
x_0 = 0, \quad y_0 = 1, \quad h = 0.1
\]
\[
\textbf{Step 2: Apply Euler’s formula.}
\]
\[
y_{n+1} = y_n + h f(x_n, y_n)
\]
\[
f(x_n, y_n) = x_n + y_n
\]
\[
\textbf{Step 3: Compute for the first step.}
\]
\[
f(x_0, y_0) = x_0 + y_0 = 0 + 1 = 1
\]
\[
y_1 = y_0 + h f(x_0, y_0) = 1 + 0.1 \cdot 1 = 1.1
\]
\[
\textbf{Result: } y(0.1) \approx 1.1
\]
\[
\textbf{Summary: Using Euler’s method, the approximate solution at } x = 0.1 \text{ is } y \approx 1.1.
\]