http://www.mapleprimes.com/posts/38888-When-Is-RungeKutta-Exact en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Sat, 13 Jun 2026 04:08:25 GMT Sat, 13 Jun 2026 04:08:25 GMT The latest comments added to the Post, When Is Runge-Kutta Exact? http://www.mapleprimes.com/images/mapleprimeswhite.jpg http://www.mapleprimes.com/posts/38888-When-Is-RungeKutta-Exact http://www.mapleprimes.com/posts/38888-When-Is-RungeKutta-Exact?ref=Feed:MaplePrimes:When Is Runge-Kutta Exact?:Comments#comment70290 <p>That seems to lead to complicated algebraic equations for higher orders, as in <a href="http://sci.tech-archive.net/Archive/sci.math.num-analysis/2007-02/msg00132.html">that post</a>?</p> <p>Alec</p> The latest comments added to the Post, When Is Runge-Kutta Exact? 70290 Thu, 07 Aug 2008 02:24:08 Z alec alec http://www.mapleprimes.com/posts/38888-When-Is-RungeKutta-Exact?ref=Feed:MaplePrimes:When Is Runge-Kutta Exact?:Comments#comment83048 <p>In that post I was looking for RK4 to give the exact solution with a particular step size.&nbsp;&nbsp; I did not dream that there were nontrivial examples where it works for all step sizes, so I didn't look for them.</p> The latest comments added to the Post, When Is Runge-Kutta Exact? 83048 Thu, 07 Aug 2008 05:34:34 Z Robert Israel Robert Israel http://www.mapleprimes.com/posts/38888-When-Is-RungeKutta-Exact?ref=Feed:MaplePrimes:When Is Runge-Kutta Exact?:Comments#comment70289 <p>Oops: some solutions seem to have been missed, namely</p> <p>a(t) = a0/(1-a0*t), b(t)=b0/(1-a0*t)</p> <p>This is the limit of the solution returned by dsolve as _C2 -&gt; infinity.&nbsp; I'll have to edit the file...</p> <p>... and here it is:</p> <p><span><a href="http://mapleoracles.maplesoft.com:8080/maplenet/primes/worksheet/4541_runge.mw">View 4541_runge.mw on MapleNet</a> or <a href="http://www.mapleprimes.com/files/4541_runge.mw">Download 4541_runge.mw</a><br /> <a href="http://www.mapleprimes.com/viewfile/2709">View file details</a></span></p> The latest comments added to the Post, When Is Runge-Kutta Exact? 70289 Thu, 07 Aug 2008 09:46:20 Z Robert Israel Robert Israel http://www.mapleprimes.com/posts/38888-When-Is-RungeKutta-Exact?ref=Feed:MaplePrimes:When Is Runge-Kutta Exact?:Comments#comment70288 <p>The solutions in this case (order 2) have form y(t)=(a+bt)/(c+dt) or y(t)=a+bt+ct^2.&nbsp;That gives a conjecture that for Runge-Kutta of order n the solutions have form y(t)=f(t)/g(t) with f(t) and g(t) being polynomials with sum of degrees less or equal than n. There may be some additional restrictions on the degree of the denominator (say, being less or equal than n/2 or floor of (n+1)/2, or something like that), because 1/f(x) with f(x) of degree 2 didn't appear in the list of solutions for Runge-Kutta of order&nbsp;2.&nbsp;</p> <p>That seems to be working for order 2n with solutions y(t)=f(t)/g(t) with f(t) and g(t) being polynomials of degree n. The differential equation in this case can be obtained by differentiating g(t)y(t)=f(t). But I didn't check all the details and I didn't check other cases.</p> <p>Alec</p> The latest comments added to the Post, When Is Runge-Kutta Exact? 70288 Thu, 07 Aug 2008 14:21:00 Z alec alec http://www.mapleprimes.com/posts/38888-When-Is-RungeKutta-Exact?ref=Feed:MaplePrimes:When Is Runge-Kutta Exact?:Comments#comment83042 <p>Aargh... still some solutions missed, because in taking _C2 -&gt; infinity you could have _C1 -&gt; infinity as well.</p> <p>Here's the latest file.</p> <p><span><a href="http://mapleoracles.maplesoft.com:8080/maplenet/primes/worksheet/4541_runge.mw">View 4541_runge.mw on MapleNet</a> or <a href="http://www.mapleprimes.com/files/4541_runge.mw">Download 4541_runge.mw</a><br /> <a href="http://www.mapleprimes.com/viewfile/2709">View file details</a></span></p> The latest comments added to the Post, When Is Runge-Kutta Exact? 83042 Fri, 08 Aug 2008 00:43:37 Z Robert Israel Robert Israel http://www.mapleprimes.com/posts/38888-When-Is-RungeKutta-Exact?ref=Feed:MaplePrimes:When Is Runge-Kutta Exact?:Comments#comment70248 <p>Interestingly, if one goes to a*y^2+b*y+c instead, then the system of equations easily implies that a(t)=0, so that if Runge-Kutta will be exact on some non-linear ODEs, one must look elsewhere than that simple generalization.</p> <p>Strangely, the system one obtains from that modification is deemed inconsistent by dsolve (via casesplit?).&nbsp; But that seems wrong - a bug?</p> The latest comments added to the Post, When Is Runge-Kutta Exact? 70248 Sat, 09 Aug 2008 18:49:05 Z JacquesC JacquesC