Solving the Differential Equation using Euler's Method

We need to approximate the value of $y(0.5)$ for the differential equation $\frac{dy}{dx} = 2 \cos\left(\frac{4\pi x}{3}\right) - y$ with the initial condition $y(0) = 1$ and step-size $h = 0.25$.

Euler's method uses the formula: $y_{n+1} = y_n + h \cdot f(x_n, y_n)$, where $f(x, y) = \frac{dy}{dx}$.

Here, $f(x, y) = 2 \cos\left(\frac{4\pi x}{3}\right) - y$. The initial values are $x_0 = 0$ and $y_0 = 1$. The step-size is $h = 0.25$. We want to find $y(0.5)$, which corresponds to $y_2$ since $0.5 = x_0 + 2h$.

Step 1: Calculate $y_1$ (approximating $y(0.25)$)

• $x_0 = 0$, $y_0 = 1$
• $h = 0.25$
• $f(x_0, y_0) = f(0, 1) = 2 \cos\left(\frac{4\pi \cdot 0}{3}\right) - 1 = 2 \cos(0) - 1 = 2(1) - 1 = 1$
• $y_1 = y_0 + h \cdot f(x_0, y_0) = 1 + 0.25 \cdot 1 = 1 + 0.25 = 1.25$

So, $y_1 \approx y(0.25) = 1.25$.

Step 2: Calculate $y_2$ (approximating $y(0.5)$)

• $x_1 = x_0 + h = 0 + 0.25 = 0.25$
• $y_1 = 1.25$
• $f(x_1, y_1) = f(0.25, 1.25) = 2 \cos\left(\frac{4\pi \cdot 0.25}{3}\right) - 1.25$
• Calculate the angle: $\frac{4\pi \cdot 0.25}{3} = \frac{\pi}{3}$
• $f(0.25, 1.25) = 2 \cos\left(\frac{\pi}{3}\right) - 1.25 = 2 \left(\frac{1}{2}\right) - 1.25 = 1 - 1.25 = -0.25$
• $y_2 = y_1 + h \cdot f(x_1, y_1) = 1.25 + 0.25 \cdot (-0.25) = 1.25 - 0.0625 = 1.1875$

Therefore, the approximate value of $y(0.5)$ is $1.1875$.