Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places.
-$$y ^ { \prime } = 1 + \frac { y } { x } , y ( 2 ) = - 3 , d x = 0.5$$
A) y1 = -4.8750, y2 = -4.0800, y3 = -4.1600
B) y1 = -3.2500, y2 = -3.4000, y3 = -3.4667
C) y1 = -6.5000, y2 = -6.8000, y3 = -13.8667
D) y1 = -6.5000, y2 = -5.1000, y3 = -6.9333

Question

Quizplus · Accepted Answer

Euler's method involves using the formula $y_{n+1} = y_n + f(x_n, y_n) \cdot \Delta x$, where $f(x, y) = 1 + \frac{y}{x}$. Starting with $y(2) = -3$ and $\Delta x = 0.5$, we calculate the first three approximations. For $y_1$, using $x_1 = 2$ and $y_1 = -3$, we get $y_1 = -3 + (1 + \frac{-3}{2}) \cdot 0.5 = -3.25$. Repeating this process for $y_2$ and $y_3$ with the updated values of $x$ and $y$, we get the sequence of approximations as given in option B.

Use Euler's Method to Calculate the First Three Approximations to the Given