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April 20th, 2016, 04:27 AM  #1 
Newbie Joined: Apr 2016 From: England Posts: 2 Thanks: 0  Rewriting second order to 1st order diff eqn
Hi I need to apply the ralston 2nd order method to find an approximate value for y(2) of the following differential equation: xy''  y' = x^2 + x With initial conditions y(1) = 1 and y'(1) = 5 So far I have done z1=y, z2=y' and z3=y'', resulting in the equation xz2' = x^2 + x + z2 This equation finds an approximate value for y'(x), but I need a value for y(x) how can I advance from this? 
April 20th, 2016, 06:00 AM  #2 
Senior Member Joined: Aug 2011 Posts: 334 Thanks: 8 
xy''y'=x^2+x Change of unknown function : z(x)=y' xz'z=x^2+x First order linear ODE easy to solve. 
April 20th, 2016, 06:45 AM  #3  
Newbie Joined: Apr 2016 From: England Posts: 2 Thanks: 0  Quote:
How would I be able to find an approximation of y(x) using that equation? Wouldn't I only be able to approximate y'(x) by using my second initial condition?  
April 22nd, 2016, 08:28 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
The point is that you have two first order equations for y and z: $\displaystyle xz' z= x^2+ x$ and $\displaystyle y'= z$. Once you have solved the first equation for z, using whatever numerical method you prefer, then do a (numerical) integration to find y. 
April 22nd, 2016, 07:14 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
[QUOTE=Don96;532996]I need to apply . . . /QUOTE] What was the original wording of the question? 

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1st, diff, eqn, order, rewriting 
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