My Math Forum Rewriting second order to 1st order diff eqn

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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
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 Originally Posted by JJacquelin xy''-y'=x^2+x Change of unknown function : z(x)=y' xz'-z=x^2+x First order linear ODE easy to solve.
JJacquelin
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: 21,030 Thanks: 2260 [QUOTE=Don96;532996]I need to apply . . . /QUOTE] What was the original wording of the question?

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