Question

Calculus Question Using Euler's Method?

In this problem we use Euler’s Method to find an approximate numerical
solution to a differential equation. Assume that y is a function of time t
which satisfies: dy/dt = y^1/3, y(0) = 10
1. Starting with y0 = 10, state Euler’s method for computing yk+1 from yk
with step size ∆t.
2. Find the values for y1, y2, y3, y4 for step sizes ∆t = 1, 0.5, 0.25 correct to 3 decimal places.
3. State all of the estimates for y(1)

Answer 1

See below.

Explanation:

After making the Euler discretization

`(dy)/(dx)=root(3)(y) ->( y_k-y_(k-1))/h=root(3)(y_(k-1))`

Beginning with `x_0=0, y_0=10` and following with `x_k=kcdot h`
we can build the successive approximations for `h=1,0.5,0.25`. Follow the plots for `h=1,0.5,0.25` in red comparing the results with the exact solution in blue which is

`y=(3 cdot 10^(2/3) + 2 x)^(3/2)/(3 sqrt[3])`