LINEAR DIFFERENTIAL EQUATIONS. 
53 
and so on down to u, which will be found to be, 
d{tanx] 
^^■^(cosx) ‘^''{co?,xy^''^^'''K^n+\')dxdxdx}; 
and if X=0, this becomes 
{y£^'^“®^'(cos cr)^y2^“"®'*'(cos 
proper precautions being taken with regard to the introduction of constants. 
Perhaps the difficulties relating to the constants may be evaded by writing the 
solution in the form 
/ d \ -("+0 
{£^^~“’'^(cos .r)^^”‘^'y£^“"^^'^(cos 57) 
and then substituting — (w+ 1 ) for n, which does not alter the original equation, we have 
{£^^“‘*’‘^(cos .r)“^y£^'*"^^'"(cos xy^d.v] . 
If a and j3 are both zero, we have for the solution of 
D%--w(w+l) — —=0, 
' ' 'cos^x ’ 
d 
“=*(j(t^) {(cos,r)-“/(cos,r)»<fo} 
where 3/= tan x. 
Let a—c and f3= — c, so that the original equation becomes 
d‘^u 
dx^ 
is 
II 
>? 
f d 
^ d 
yd (tan x) 
)-V= 
'cos"' a? 
71+1 
which contains the proper number of constants ; as the constant which enters by 
reason of the first integration disappears by the subsequent differentiations. 
d 
This solution will apply to Laplace’s equation, if for c be written c^* 
This gives for the solution of 
d’^u JPu , “ 
djc^ ^ dy )cos^a;’ 
«— {£-"'5(008 x)-“f(cos xy‘(p(y+2cx)dx} 
{(cos*)-“'x(</-2cx)}, 
