Mathematical Tools for Physics

4—Differential Equations 97

x

t′

impulse

t

When the external force is the sum of two terms, the total solution is the sum of the solutions for the
individual forces.
m ̈x+kx=F 0 +F 1 (22)

If the two terms are two steps

F 0 =

{

F(t 0 ) (t 0 < t < t 1 )

0 (elsewhere)

and F 1 =

{

F(t 1 ) (t 1 < t < t 2 )

0 (elsewhere)

then ifx 0 is the solution to Eq. ( 22 ) with only theF 0 on the right, andx 1 is the solution with onlyF 1 , then the
full solution to Eq. ( 22 ) is the sum,x 0 +x 1.
Think of a general forcing functionFx,ext(t)in the way that you would set up an integral. Approximate it
as a sequence of short steps as in the picture. Betweentkandtk+1the force is essentiallyF(tk). The response
ofmto this piece of the total force is then

xk(t) =

{F(t

k)∆tk

mω 0 sin

(

ω 0 (t−tk)

)

(t > tk)

0 (t≤tk)

where∆tk=tk+1−tk.

F

t (^1) t 2 t^5
t