42, 43] DAMPED OSCILLATIONS. 149 



This equation is linear with constant coefficients, and is a type of 

 those that appear in the theory of oscillations. The fundamental 

 property of such equations is that any solution multiplied by a 

 constant is a solution, and that the sum of two solutions is a 

 solution. In order to find a particular solution we put 



s = e lt . 

 Differentiating we have 



Substituting in the differential equation we may divide out the 

 factor e**, obtaining 



27) A 2 + x* + W = 0, 



a quadratic to determine the constant >L. 

 Calling its roots A 1; 1 2 we have 



28) ^ = -1 



The general solution is obtained by multiplying the- particular solu- 

 tions e* 1 * and e^ by arbitrary constants and adding. Thus we obtain 



29) s 



We have to consider two cases, 



I. K 2 > 

 II. % 2 < 



In case I the radical is real, and since its absolute value is less 

 than ;c both Aj_ and ^ are negative and s eventually decreases as the 

 time goes on, vanishing when t = oo. We have 



This vanishes when 



or 



Consequently if 5 and ^i are of opposite signs s will increase to a 

 maximum and then continually die away. If they are of the same 

 sign the motion dies away from the start. Both cases are shown in 

 Fig. 31, where t is the abscissa and s the coordinate. 



In case II the radical is imaginary and both ^ and A 2 are 

 complex. Then writing 



