Math 234

Pre-Lab 8: Initial Value Problems (Sec 4.3, 4.5)

To solve an Initial Value Problem using Laplace Transforms...

i.) Take the Laplace Transform of both sides of the DE.

ii.) Solve the resulting algebraic equation for Y(s).

iii.) Take the Inverse Laplace Transform to obtain y(t).

We use the following rules for the Laplace Transforms of ����. ���� � Rule 35: ���� � � � ��0� Rule 36: ���

� � �� � ���0� � �′�0�

1.) Solve the Initial Value Problem below using Laplace Transforms.

� � 3� � 4��� ,��0� � 12

As a quick check, plug t=0 into your answer to verify ��0� � 12.

The unit step (Heaviside) function is defined as

��� � �� � �0if� < �1if� ≥ � Its Laplace Transform is

Rule 25: ����� � ��� � !"#$

2.) Graph %��� � 2��� � 1� � 2��� � 3�. Then find its Laplace Transform.

If we multiply a function by ��� � ��, it turns that function off until time t=c.

�� ��� � 3���

The Laplace Transform of a function times ��� � �� is Rule 28: ����� � ��'���� � �()$��'�� � ���

Ex Find ����� � 3����. Matching to Rule 28, if we set � � 3 and '��� � �� we get ����� � 3�'���� � �(*$��'�� � 3��

� �(*$���� � 3��� � �(*$���� � 6� � 9� � �(*$ - 2�* �6�� �

9�.

3.) Find ����� � 5�����. (Hint: �0�12 � �0��2 and �2 is just a constant.)

If you see an exponential function �()$ in the frequency domain, then the Inverse Laplace Transform will result in a unit step function.

Rule 27: �(3��()$4���� � ��� � ��%�� � ��

Ex Find �(3 � !5#$6 7. Matching to Rule 27, let � � 8 and 4��� � 3$6. Then %��� � �(3 � 3$67 �

39�(3 �

*!$; !=?#$@1A 7.

The unit impulse (Dirac delta) function is defined as

B�� � �� � �1if� � �0otherwise This function is useful for representing instantaneous actions.

Its Laplace Transform is

Rule 26: ��B�� � ��� � �()$

6.) Machop attaches a 1kg punching bag to the end of the spring with

spring constant k=9. There is no damping force acting on the spring.

Machop holds the bag at rest position and releases it. At 10 seconds,

Machop gets bored and punches the bag, applying an instantaneous

downward force of 4 Newtons.

a.) Recall the basic spring equation is: J�

� K� � L� � 4���. Fill in the values in the DE and the initial conditions.

_____ �

�_____ � �_____ � = _______________ , ��0� �______, ��0� �_____

b.) Solve the Initial Value Problem. (Hint: Use your answer to #5.)