Question

Consider IVP: y' = 9x -y ; y(0)=9
a) Write down Euler's method for yn+1 in terms of yn for a general stepsize.
b)Using one step of Euler's Method approximate y(1).
c) Using three steps of Euler's Method approximate y(1).

Answer 1

Euler's method uses the following recurrence to generate an approximation of the solution to an ordinary differential equation \(y'=f(x,y)\):
\[y_{n+1}=y_n+hf(x_n,y_n)\]where \(h=x_{i+1}-x_i\) is the step size and the ODE is \(f(x,y)=9x-y\)

It should be easy enough to see that the formula in this case becomes
\[y_{n+1}=y_n+h(9x_n-y_n)=(1-h)y_n+9hx_n\]So for example, given that \(y(0)=9\), and you wanted to approximate \(y\left(\dfrac{1}{2}\right)\) within two steps (so \(h=\dfrac{1}{4}\)), that would generate a table of intermediate computations like this:
\[\begin{array}{c|c|c|c|c}
n&x_n&y_n&f(x_n,y_n)&y_{n+1}\\[1ex]
\hline
0&0&9&-9&\dfrac{27}{4}\\[1ex]
1&\dfrac{1}{4}&\dfrac{27}{4}&-\dfrac{9}{2}&\dfrac{45}{8}
\end{array}\]and so \(y\left(\dfrac{1}{2}\right)\approx\dfrac{45}{8}\).

Answer 2

Using three steps of Euler's Method approximate y(1).