DIFFERENTIAL EQUATIONS 
343 
EXERCISES 
1. Find one solution of the differential equation 
^_3^ + 3^_ v = 0 
dx 3 dx 2 dx 
by the method of § 12, and prove by direct substitution that xe x and 
x 2 e x are also solutions. What is the general solution? 
Ans. y — (c 0 + c x x + c 2 x 2 )e x . 
2. Find two solutions of the differential equation 
dh/ 
da 4 
+ 2 
dry 
dx 2 
+ y = 0 
by the method of § 12, and prove by direct substitution that x sin x 
and x cos x are also solutions. What is the general solution ? 
Ans. y = (a + bx) cos x + (c + dx) sin x. 
Solve completely the following 
3. 
d 4 y . d 3 y _ q 
dx 4 d.« 3 
differential equations. 
4 tfy , d^ =0 
dx 4 da; 2 
5. 
dæ 3 da; 2 da; 
6. 
2^ + v = 0. 
da; 6 da; 3 
14. Small Oscillations of a System with n Degrees of Freedom. 
We treat here only that part of the problem which relates to the 
integration of the differential equations involved.* Let the kinetic 
energy, T, and the force-function, U, be given by the equations : 
t = X a i i q* ti + T » U =~X bi > qi q > + Uu { 
The two quadratic forms are both definite, and a u , b {j are constants- 
Lagrange’s Equations : 
_dJT = dJJ 
dt dq[ dq k dq k 
give: 
(1) «¡tiQ'l' -4 h UknQn = — (fikiQi H + b kn q n ). 
By means of a suitable linear transformation, 
(2) q k = p-aPi + ••• + y-knPn) k = l,-”,n. 
the two quadratic forms can be reduced to the normal form : f 
* Cf. Appell, Mécanique rationelle, voi. 2, p. 343. 
t Bòcher, Algebra, Chap. 13.