DERIVED SOLUTIONS OF DIFFERENTIAL EQUATIONS. 



(Short Methods). 



R. D. BOHANNAN. 



When F(m)=0 has n roots equal a, then F(m)=0 and its 

 first n — 1 derivatives have this root in common. So if F(m, x) 

 is a solution of a differential equation, for different values of m, 

 then, if n values of m are equal to a, are also the first n — 1 

 partial derivatives of F(m, x) with respect to m also, generally, 

 solutions, when, after differentiation, m is changed to a. 



Case I. 



(a) Linear differential equations with constant coefficients, 

 second member zero. 



y = gmx 



is the solution. 



If (D — a)" is a factor of the first member, using the symbolic 

 method of solution, then 



pinx vpinx v2pmx •vn— Ipmx 



are all solutions, when m is changed to a, these being the partial 

 derivatives with respect to m. Multiply each of these by an 

 arbitrary constant and add for the general solution correspond- 

 ing to (D — a)". 



(b) When D^+a- is a factor of the first member, 



y = sin mx, y = cos mx 



are solutions, when m is a. 



When (D- + a-)" is a factor of the first member, 



sin ax, x sin (7r/2 + ax), x^sin (27r/2 + ax), 



x'^-i sin ( (n — 1) 7r/2H-ax), 



cos ax, X cos {ir/2-\-a,x, x^cos (27r/2 + ax), 



x"-i cos ( (n — 1) 7r/2 + ax) 



are solutions. 



(c) When (D — a)2+b- is a factor of the first member, 



y ^gmx 



is a solution, when m is changed to a+bi and a — bi, giving, 



y = e''-^ cosbx and y = e^^ sin bx 

 as solutions, after addition and subtraction. 



231 



