Calculus Question Using Euler's Method?

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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)

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Answer 1 · 2017-02-09T15:28:05

See below.

Explanation:

After making the Euler discretization

$\frac{d y}{d x} = \sqrt[3]{y} \to \frac{y_{k} - y_{k - 1}}{h} = \sqrt[3]{y_{k - 1}}$ $(dy)/(dx)=root(3)(y) ->( y_k-y_(k-1))/h=root(3)(y_(k-1))$

Beginning with $x_{0} = 0, y_{0} = 10$ $x_0=0, y_0=10$ and following with $x_{k} = k \cdot h$ $x_k=kcdot h$
we can build the successive approximations for $h = 1, 0.5, 0.25$ $h=1,0.5,0.25$ . Follow the plots for $h = 1, 0.5, 0.25$ $h=1,0.5,0.25$ in red comparing the results with the exact solution in blue which is

$y = \frac{{(3 \cdot 10^{\frac{2}{3}} + 2 x)}^{\frac{3}{2}}}{3 \sqrt{3}}$ $y=(3 cdot 10^(2/3) + 2 x)^(3/2)/(3 sqrt[3])$

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Calculus Question Using Euler's Method?

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