342 
CALCULUS 
20. Solve the differential equation (cf. Ex. 11, § 11): 
dx* clx 
Ans. y = C x x? + C 2 xt. 
13. Continuation. Equal Roots.* If equation (2) of § 12 has 
equal roots, then each is equal to — a, and y x = e~ ax is a solution of 
the given differential equation. 
We can guess a second solution by considering the case that the 
roots are not quite equal, one being — a and the other — a -f- h. 
Let 
y __ Q—a.x-\-hx 
Then 
V -Vl_ ,-ax ~ 1 
h ~ h 
is a second solution of the near-equation, no matter how small li be 
taken. Now, this function is nearly equal to xe~ ax when li is small. 
So we are led to try this function, and it turns out on substituting it 
that it does satisfy the given differential equation. 
Thus we find as the general solution 
y = e~ ax (.A + Bx). 
If n > 2, equation (9) of § 12 may have more than two roots equal. 
It is not hard now to guess by analogy what the solutions will be in 
this case. If m be an Z-fold real root, then 
yi = e mx , y 2 = xe mx , • • •, y t = x l ~ l e mx 
will be l linearly independent solutions. If, on the other hand, m is 
imaginary: m = p + qy/ — 1, then p — gV — 1 will also be a root, 
and we have each of these roots counting l times. The functions 
y 2k+1 = x k e px cos qx, y 2k+ o = x k e px sin qx, Ic = 0,1, •••, l — 1, 
are here 21 linearly independent solutions. 
The case that the m-equation for a simultaneous system of the type 
of § 12, Exs. 17 and 19, has equal roots is more complex ; cf. Goursat, 
Cours d’analyse, Vol. II, 2d ed. (1911), Chap. XX, § 420, p. 483. 
* This case is unimportant in practice ; and yet it is necessary to treat it if 
the theory is to be complete. The student may safely postpone this paragraph 
till he needs to use it.