DIFFERENTIAL EQUATIONS 
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equation (2) whenever one single particular solution of (1) can be 
found, as is stated in precise form by the following 
Theorem. Let Y be a particular solution of (1); i.e. a function 
which satisfies (1), hut contains no constants of integration. Then the 
general solution of (1) is 
y= Y+c x y x + ••• + c n y n , 
where y x , • ••, y n are n linearly independent solutions of the correspond 
ing homogeneous differential equation (2), and c u •••, c n are arbitrary 
constants. 
By hypothesis, we have the equation 
Now let y he any solution whatever of (1). Subtract (7) from 
(1); then 
i.e. the function y — Y satisfies (2). It can, therefore, he written in 
the form (6): 
y - Y= c x y x + c 2 y 2 + ”• + c n y n , 
and the theorem is proved. 
EXERCISES 
Show that the following functions are linearly independent when 
1. since, coscc. 
3.* x, e x . 
2. e~ kt sin pi, e~ kt cos pt. 
4. e x , sin x. 
6. e mx , xe mx . 
5. e px , . e qx , p^q. 6. e mx , xe mx . 
Are the following functions linearly independent ? 
7. n = 3: e ax , e* 3 * cospx, e? x sin px. 
8. n = 4: sin px, cos px, sin qx, cos qx. 
9. n = 4 : e ax sin px, e“* cos px, e? x sin qx, e& x cos qx. 
10. By a simultaneous system of n linear differential equations of the 
first order is meant: 
* Suggestion. Assume the theorem false. Then Ax -f Be x = 0 for all values 
of x ; and now differentiate.