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ODE analytical solution

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John
John 2011-3-26
How can I find the analytical solution to the following ode?
y'+ x/y=0
when x=0 and y=5.

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bym
bym 2011-3-26
simplify(dsolve('Dy=-x/y','y(0)=5','x'))
ans =
(25 - x^2)^(1/2)
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John
John 2011-3-27
From my text book I have coded Euler's Method to solve ODEs but i am confused on where to input each into the program the code I have is
function [t,y] = eulode(dydt, tspan, y0, h)
%eulode: Euler ODE solver
% [t,y] = eulode(dydt, tspan, y0, h, p1, p2,...)
% ` uses EULER'S method to INTEGRATE an ODE
% (uses the slope at the beginning of the stepsize to graph the
% function.)
%Input:
% dydt = name of hte M-file that evaluates the ODE
% tspan = [ti,tf] where ti and tf = initial and final values of
% independent variables
% y0 = initial value of dependent variable
% h = step size
% p1,p2 = additional parameter used by dydt
%Output:
% t = vector of independent variable
% y = vector of solution for dependent variable
if nargin<4, error('at least 4 input arguments required'), end
ti = tspan(1); tf = tspan(2);
if ~ (tf>ti), error('upper limit must be greater than lower limit'), end
t = (ti:h:tf)';
n = length(t);
%if necessary, add an additional value of t
%so that range goes from t=ti to tf
if t(n)<tf
t(n+1) = tf;
n = n+1;
t(n)=tf;
end
y = y0*ones(n,1); %preallocate y to improve efficiency
for i = 1:n-1 %implement Euler's Method
y(i+1) = y(i) + dydt(t(i),y(i))*(t(i+1)-t(i));
end
plot(t,y)
in my other m-file I am supposed to run the program with 3 different step sizes (.5; .1; .01) so my code looks like this
dydx=@(x,y) -(x/y);
[x1,y1]=eulode(dydx, [0 1],5,.5);
[x2,y2]=eulode(dydx,[0 1],5,.1);
[x3,y3]=eulode(dydx,[0 1],5,.01);
disp([x1,y1])
disp([x2,y2])
disp([x3,y3])
I think this is correct because each answer is displaying a value close to 5 (which the analytical value should be 5 since x=0)
Did i code this right?

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