DIFFERENTIAL EQUATIONS 167 



8.062 The equation: 



X(P), 

 may be reduced to 8.061 by dividing by \l/(p). 



DIFFERENTIAL EQUATIONS OF AN ORDER HIGHER THAN THE FIRST 



8.100 Linear equations with constant coefficients. General form: 



d n y d n ~ 1 y d n ~ 2 y 



d^ + ai d^ + a *d^ + - +*->&). 



The complete solution consists of the sum of 



(a) The complementary function, obtained by solving the equation with 

 V(x) = o, and containing n arbitrary constants, and 



(b) The particular integral, with no arbitrary constants. 



8.101 The complementary function. Assume y = e^ x . The equation for 

 determining X is: 



X n + ffiX"- 1 + a 2 \ n - 2 + + a n = o. 



8.102 If the roots of 8.101 are all real and distinct the complementary function 

 is: 



y = c\e** x + cie^ x + .... + c n e^n x . 



8.103 For a pair of complex roots: 



ju iv, 

 the corresponding terms in the complementary function are: 



e x (A cos vx + B cos vx) = Ce x cos (vx - 6) = Ce x sin (vx + 6), 

 where 



C = VA 2 + B 2 , tan0 = ^- 



A. 



8.104 If there are r equal real roots the terms in the complementary function 

 corresponding to them are: 



where X is the repeated root, and AI, A 2 , , A r are the r arbitrary constants. 



,8.105 If there are m equal pairs of complex roots the terms in the complementary 

 function corresponding to them are: 



e x { (Ai + A 2 x + A 3 x 2 + .... + A m x m ~ 1 } cos vx 



+ (Bi + B 2 x + B 5 x 2 + ....+ B m x m ~ 1 } sin vx} 



= e x {d cos (vx - 61) + C 2 x cos (vx - 2 ) + + C m x m ~ l cos (vx - TO )} 



= e^ x {Ci sin (vx + 0i) + C 2 x sin (vx + 2 ) 4- + C m x m ~ l sin (vx + m )} 



