350 



Also y = kef is a solution of the equation -3-^ = n 2 x if 

 a 2 = n 2 or a = m, 



and y = Ae ina: + Be -ina! is the complete solution 

 but y = A(cos nx + i sin nx) + B(cos nx i sin nx) 

 =-= (A + B) cos nx + i(A. B) sin nx 

 = C cos nx -f D sin nx where C and D are constants 



177. The general solution y = kef can also be assumed for the 

 differential equation -5-^ + ^ a ~r- + b' 2 y = 0. 

 For if y = ke? x , 



dx 



Then +2a-+ b*y - a 2 ^^ + 2a 



tttC CkC 



= fce^a 2 + 2aa + b 2 ) 



= 0, if a 2 + 2aa + & 2 = 



($"U tl$J 



Hence ?/ = kef will be a solution of the equation -^ + 2a~- 



dx dx 



+ b 2 y 0, providing a has the values which will satisfy the 

 quadratic equation a 2 + 2aa + b 2 = 0. 



Let a x and <x 2 be the two roots of this equation. 



Then y = A(? lX + Be 02 * will be the complete solution of the 

 differential equation. 



For since o^ and <x 2 are the roots of the equation a 2 + 2aa 

 + b 2 = 0. 



Then aj + 2a^ + b 2 = 



and a 2 , + 2aa 2 + & 2 - 



If ?/ = A^i* + Be **, 



-^ = ajAe ! 35 + BaoeV 



ajA^i 85 + y^e?** + 2a(a 1 Ae fl i a! + 



