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Differential Equations of Motion
Differential Equations of Motion
A differential equation for y.t/ can involve dy=dt and also d 2 y=dt 2
Here are examples with solutions C and D can be any numbers
d 2y
d 2y
yD C cos t C D sin t
D �y and
D �!2 y Solutions
2
yD C cos !t C D sin !t
dt
dt 2
Now include dy=dt and look for a solution method
dy
dy
d 2y
C 2r
C ky D 0 has a damping term 2r : Try y D e t
2
dt
dt
dt
Substituting e t gives
m2 e t C 2re t C ke t D 0
m
Cancel e t to leave the key equation for m2 C 2 r C k D 0
r 2 � km
Two solutions 1 and 2
m
The differential equation is solved by y D Ce 1 t C De 2 t
The quadratic formula gives D
�r �
�
Special case r 2 D km has 1 D 2
EXAMPLE 1
1 ; 2
�r �
D
Solution
d 2y
dy
C 6 C 8y D 0
2
dt
dt
m D 1 and 2r D 6 and k D 8
�
�
r 2 � km
is �3 � 9 � 8
m
y D Ce �2 t C De �4 t
EXAMPLE 2
Then t enters y D C e 1 t C Dte 1 t
Then
1 D �2
2 D �4
Overdamping with no oscillation
�
Change to k D 10 D �3 � 9 � 10 has
Oscillations from the imaginary part of 1 D �3 C i
2 D �3 � i
Decay from the real part �3
Solution y D C e 1 t C De 2 t D C e .�3Ci /t C De .�3�i /t
e i t D cos t C i sin t leads to y D .C C D/e �3t cos t C .C � D/e �3t sin t
EXAMPLE 3
Solution
Change to k D 9 Now D �3; �3 (repeated root)
y D C e �3t C Dte �3t includes the factor t
2
Differential Equations of Motion
Practice Questions
1. For
d 2y
D 4y find two solutions y D C e at C De bt : What are a and b ?
dt 2
2. For
d 2 y
D �4y find two solutions y D C cos !t C D sin !t: What is ! ?
dt 2
3. For
d 2 y
D 0y find two solutions y D C e 0t and (???)
dt 2
4. Put y D e t into 2
5. Put y D e t into 2
6. Put y D e t into
d 2y
dy
C 3 C y D 0 to find 1 and 2 (real numbers)
dt
dt 2
d 2y
dy
C 5 C 3y D 0 to find 1 and 2 (complex numbers)
dt
dt 2
d 2y
dy
C 2 C y D 0 to find 1 and 2 (equal numbers)
2
dt
dt
Now y D C e 1 t C Dte 1 t : The factor t appears when 1 D 2