DIFFERENTIAL EQUATIONS 
341 
15. Show that one solution of the differential equation 
d n y . d n ~ 1 y , 
â^ +a 'd^ + "• + “" y=e ’ 
where oq, • ••, a n , e are constants, and a n =£ 0, is the function y = e/a n . 
16. Obtain by inspection one solution of the differential equation 
<&y + 2^= 4 
dx 2 dx ’ 
and hence solve the equation completely. 
17. In the case of a simultaneous system of homogeneous linear 
differential equations with constant coefficients, as 
§l = Ay + Bz, -j- = Cy + Dz, 
dx dx 
it is reasonable to try for a solution of the form: 
y = \e mx , z = ¡xe mx . 
Show that two such solutions can be found if the equation 
A — m B 
= 0 
6 D — m 
has two distinct real roots, and determine the ratio, X/y.. 
Apply your results to the case : 
dx 
= 6y- 42!, 
dz o 
— = 3 y-z. 
dx 
Ans. The complete solution is : 
y = C^e 2 * + 4:C 2 e 3x , 
z = C^ + SCté*. 
18. Develop the theory for the case that the quadratic in m, 
Question 17, has imaginary roots. 
19. Extend Questions 17,18 to the case of a system of three equa 
tions, 
— = A\U + B^v + CiW, 
dx 
—— = A% u -(- B 3 v -f- C 3 w, 
dx 
~ = A 3 u + B 3 v + C 3 w. 
dx 
Hence generalize to the case of n equations.