DIFFERENTIAL EQUATIONS 



173 





8.232 The particular integral. If A* is the minor of i~ in A, the par- 

 ticular integral is: 



8.233 If yi is one integral of the equation 8.230 with v = o, the substitution 



du 



will result in a linear equation of order n - i. 



8.234 If y\, yz, , y n -i are n i independent integrals of 8.230 with 



V = o the complete solution is: 



.j ^j y k j ~s? e p x dx 



where A is the determinant: 



n i 



y = /j Ckk 



x n ~ 2 dx n 



dx n ~ 3 dx n ~ 3 ' 



dx dx 



dy n -i 

 dx 



y n -i 



Jn 2/y, 



and Afc is the minor of -, r- in A. 



dx n ~ 2 



SYMBOLIC METHODS 



8.240 Denote the operators: 



T- D 



dx 



*-e 



A J t/. 



dx 



8.241 If X is a function of x: 



i. (D -m)- l X= e mx f e~> 



Xdx. 



(0 -m}' l X= x m j x~ m ~ l 



Xdx. 



(0 _ w )-i 



