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}`