Question

Use Euler's method with step size 0.5 to compute the approximate y-values y y(1.5), y2y (2), yzy (2.5), and y4 y(3) of the solution of the initial-value problem y' = 2 – 5x + 2y, y(1) = 0.

Answer 1

Let $f(x,y)=y'$. In Euler's method, we start with an initial value $y_0=y(x_0)$ and recursively compute

$\begin{cases}x_{n+1}=x_n+h\\y_{n+1}=y_n+f(x_n,y_n)h\end{cases}$

for $n\ge0$ and where $h=0.5$.

For example, when $n=0$, we have

$y_1=y_0+0.5(2-5x_0+2y_0)\implies y_1=0+0.5(2-5+0)\implies y_1=-1.5$

and so on. A table can help organize this:

$\begin{array}{ccccc}n&x_n&y_n&f(x_n,y_n)&y_{n+1}\\0&1&0&-3&\boxed{-1.5}\\1&1.5&-1.5&-8.5&\boxed{-5.75}\\2&2&-5.75&-19.5&\boxed{-15.5}\\3&2.5&-15.5&-41.5&\boxed{-36.25}\end{array}$