Differential equations, euler's method and bisection method

[meddi83]

Homework Statement

Hi guys, I have the following problem and I don't know how to start.

I am given that W = 0.5, X(0) = 0, Iab = 20m and ha = 5m $$\frac{dy}{dx}=0$$

and

$$\frac{dy^2}{d^2x}=\frac{W}{T}\sqrt{1 + (\frac{dy}{dx})^2}$$

I am told to convert the 2nd order ODE to two 1st order ODEs at first. Then I am asked to choose an arbitrary value of T and integrate the system of equations using Euler's method from 0-Iab, where Iab=20, with 3 different step sizes. Then, I have to select the step size that gives an accurate value of y(Iab) (i.e. approximate relative error less than 0.1% if comparing with ha = 5).. Then I am asked to chose two values of T such that one gives y(Iab) smaller than ha and the other one gives y(Iab) bigger than ha and use those two values with the bisection method

Homework Equations

Euler's method: $$yi+1 = yi + f(xi,yi)*h$$ where $$f(x,y) = \frac{dy}{dx}$$

The Attempt at a Solution

At first I broke it to two 1st order ODES:

$$\frac{dy}{dx}=g$$ and $$\frac{dg}{dx}=\frac{W}{T}\sqrt{1 + g^2}$$

Then, for Euler's method: $$yi+1 = yi + gi*h$$ and $$gi+1 = gi + (\frac{W}{T}\sqrt{1+gi^2})*h$$

My problem is, I don't know where to start in order to find y(Iab) (because the problem is asking for an accurate value of y(Iab) )... and what step size (h) to use. .. I also don't know which equation shall be used with the bisection method..

any help would be really appreciated. thank you

 

[HallsofIvy]

I would suggest you go back and reread the problem. You say that X(0)= 0 but your differential equation has y as a function of x. What does X(0) mean?

 

[meddi83]

Sorry, typo mistake. It says that the slope, at x=0, is dy/dx=0

 

[meddi83]

and from the Picture I have, the graph, when x=0 -> y=0.

So I suppose for euler's method, suppose I choose a step size of ie, h=5, then it would be

$$y(5) = y(0) + g(0)*h$$ and $$g(5) = g(0) + (\frac{W}{T}\sqrt{1+g(0)^2})*h$$

Is that correct ? And since I have to integrate from 0-20, I suppose I'll have to do y(10), y(15) and y(20) aswell, right ? (Same for g(10),g(15),g(20) ) ?