276 Integration of Differential Equations. 



not stop to give it, but shall refer for the process to the same vol- 

 ume and page of Lacroix's work that we have done at the com- 

 mencement of this paper. 



We will now show how to find the integral of (2). If we 

 change h into 6\/ — 1, in (6), we shall have the complete value of 

 y in the general case when p is positive and less than unity ; and 

 if p=^, c=l, by making the same change in (e) we shall have 

 the complete value of y in this case of the general integral. 



If p is positive and greater than unity, we must change h into 

 iV — 1 in Lacroix's integral, and change —v^ in ours into -\-v^y 

 then using A and B to denote two arbitrary constants, we get 



fa p-\ /»infin. \-c -p 



y=A./ dx{a^ —x^) cos.6w%v/ — 1+B/ u dv{a^'\-v^) 



cos. hu'^v, (/), for the complete value of y. If J5 is negative and 

 finite, we must change —x^ into -\-x^ in Lacroix's integral, and 

 h into hV — 1 in ours, then using A and B for the arbitrary con- 



stants we shall ha.vey = A/ dx{a^-\-x^) cos.6M%+Bit 



dv{a^ —v^) cos.bu^v\/ — 1, (g), for the complete value of y. 



We will now give some applications of what has been done to 

 differential equations which can be reduced to the form of (2). 



d^y Ady 

 Suppose that we would integrate the equations 'J~^'i-V2x'~ 



d^y dy . 



B2a;"y=0, (h), and^ -f A^-B2e"'y=0, (i), in which x and 



y are the only variables, y being considered as a function of x, 

 X being considered as the independent variable whose differential 

 dx is supposed to be constant, and e is supposed to be the base of 

 hyperbolic logarithms ; if we change the independent variable 

 from X to u^ we must not consider </.i; to be constant, but we may 

 suppose du to be constant ; also in (A,) and (z) we must suppose 



d^v \dx I 



Ihat-F^ is changed to — -j — We shall now put M = a;''+2j 



n+2=m, and du= constant in {h\ then since y is regarded as a 

 function of u, and u of x, we get by well known formulas of 



dy dy du dy \dxi 



differentiation ^- = t- if' = '^ 17, ^"""'j and hence —^ — ~ m 



A 



