An Algorithm for Solving Second Order Linear Homogeneous Differential Equations

In th is paper we present an algor i thm for f inding a "c losed-form" solut ion of the dil l-erential equatiofl -1"'+ ql +by, where a and b are rational functions of a complex var iable .x. provided a "c losed-form" solut ion exists. The algor i thm is so arranged that i f no solut icrn is found. then no solut ion can exist . The frrst sect ion makes precise what is meant by "c losed-form" and shows that there are four possible cases. The f i rst three cases are discussed in sect ions 3. 4 and 5 respect ively. The last case is the case in which the given equat ion has no "c losed-form" solut ion. I t holds precisely when the f i rst three cases fai l . In the second sect ion we present condi t ions that are necessary for each of the three cases. Al though this mater ia l could have been omit ted. i t seenas desirable to know in advancc which cases are possible. The algor i thm in cases I and 2 is qui te s imple and can usual ly be carr ied out by hand. provided the given equat ion is relat ively s imple. However. the algor i thm in case 3 invt l lves qui te extensive computat ions. I t can be programmed on a computer for a spepi f ic di f ferent ia l equat ion wi th no di f l icul ty, In fact . the author has Worked through ;everal cxamples using only a programmable calculator. Only in one example was a coinputer necessary. and this was because intermediate numbers grew.to 20 decimal digi ts. more than the calculator could handle. Fortunately, the necessary condi t ions for case 3 are quite strong so this case can often be eliminated from consideration. The algor i thm does require that the part ia l f ract ion expansion of the coeff ic ients of the di f ferent ia l equat ion be known. thus one needs to factor a polynomial in one var iable over the complex numbers into l inear factors. Once the part ia l f ract ion expansions are knou'n. only l inear algebra is required. Using the MACSYMA computer algebra system. see. for example. Pavel le & Wang (1985). Bob Caviness and David Saunders of Rensselear Polytechnic Inst i tute programmed the ent i re algor i thm (see Saunders (1981)) . Meanwhi le. the algor i thm has