1 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 2y D �4y ﬁnd two solutions y D C cos !t C D sin !t: What is ! ? dt 2 3. For d 2y 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

Differential Equations of Motion

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