DIFFERENTIAL EQUATIONS 183 



8.521 The function F(p, q, ) can always be found if it is known for positive 

 proper fractional values of p and q. 



8.522 p and q positive improper fractions: 



p = m + r, q = n + s 

 where m and n are positive integers and r and 5 positive proper fractions. 



8.523 p and q both negative: 



p = - ( m - i + r) q'= - (n - i + s), 





8.524 p positive, q negative: 



p = m + r, q = - n + s, 



F(m + r, - n + s, Q = ^ |+i-r-* Jl p(i - s, i - r, ) 1 



8.525 p negative, q positive: 



p = -m + r, q = n + s, 



8.530 If either p or q is zero the relation D = o is satisfied and the complete 

 solution of the differential equation is given in 8.502, 3. 



8.531 If p = m, a positive integer: 



* = F(m, q, Q - Cl -< 



8.532 If _/> = m, a positive integer and both q and are positive: 

 <i> = F(m, q, ) = ci I *u m - l (i - u) q ~ l e~^ u du + c z e~^ I (i + w)" 1 " 1 ^" 1 



8.533 If q = w, a positive integer: 



, , Q = C1 H 



8.534 If ^ = n, a positive integer and both p and are positive: 

 $ = F(p, n, Q = ci fW-?(x - w)^- 1 e-^ w ^ + c 2 e-^ / (i + u)?- l u n 



Jo t/o 



