Euler's Method Assignment.sagews

Euler's Method Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Question 1

Use Euler's Method to graph an approximate solution curve to $\displaystyle\frac{dy}{dx}=x^2-y^2$ with initial value $(1,2)$. Graph on the interval from $x=1$ to $x=3$ and use $n=20$ steps.

Question 2

Consider the initial value problem $\displaystyle\frac{dy}{dx}=6x^2-3x^2y\quad y(0)=3$.

Part a

Use Euler's Method with $n=50$ steps to graph an approximate solution curve on the interval from $x=0$ to $x=1$.

Part b

The exact solution of this differential equation is $y=2+e^{-x^3}$. Add a graph of this curve to your graph in part a.

Question 3

Consider the initial value problem $\displaystyle\frac{dy}{dx}=y^2\cos(x),\quad y(0)=0.5$.

Part a

Approximate $y\left(\frac{\pi}{6}\right)$ using Euler's Method with $n=20$.

Part b

Use desolve to find the exact solution of the original initial value problem.

[You'll have to solve for $y$, either by hand or use Sage.]

Part c

Using the solution from part b, find the exact value of $y\left(\frac{\pi}{6}\right)$.

Part d

Subtract your approximation (part a) from the exact value (part c). This is the error.

Part e

Approximate $y\left(\frac{\pi}{6}\right)$ using Euler's Method with $n=50$.

Part f

Calculate the new error. [This should be smaller than the error in part d.]

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.