MTH 304 Final 2010

Lebanese American University Diff. Equations Fall 2010 Byblos Final Exam 2 Hours Name : ------------------------------------------------------------------------- - - - - - - - - - - - - - - - - - - - - 1. Consider the linear system � x� � 4 x � y y� � x � 4 y a) Find the general solution to the system, and determine the type and stability of the critical point O. b) Draw the phase portrait. (15 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2. Determine the type of the critical points of the nonlinear system � x� � 2 x � y � � �2 (12 pts) 2. Determine the type of the critical points of the nonlinear system � x� � 2 x � y y� � �y � x�2 � y � x (12 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 3. Using the undetermined coeficients method, solve the 2ndorder LDE : y� � 3 y� � 2 y � x e2 x (12 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4. Solve the LDE: x2 y� � x y� � y � x2 sin x where x � 0 (12 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 5. Solve, by 2 methods, the IVP : y� � x2 y � y x , y�1� � 3 1st method : Change the Bernoulli equation to a LDE, then solve. 2nd method : Use the Improved-Euler's method. Perform 2 steps with h � 0.2 (15 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Untitled-1 1 Turn Over - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 6. Using Laplace transforms, solve the IVP: � y� � y� � r�t� y�0� � 0, y��0� � 2 where r �t� � � 1 if t � 2 t if t � 2 (14 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 7. A mass of 2 Kg is attached to the lower end of a spring with spring constant k � 6. We pull down the mass 0.1 m and release it an upward initial velocity of 4 m/sec. Using Laplace transforms, find the position of the mass at any time t. ( 8 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 8. Using Laplace transforms, solve the system of LDE: � x� � x � 8 y y� � x � 3 y where x�0� � 2, y�0� � 1 (12 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Formulas: (1) £ �y�� � s £ �y� y�0� (2) £ �y�� � s2 £ �y� s y�0� y��0� Untitled-1 2 Formulas: (1) £ �y�� � s £ �y� � y�0� (2) £ �y�� � s2 £ �y� � s y�0� � y��0� (3) £ �eat f�t�� � F�s � a� (4) £ �� 0 t f�x��x� � 1 s £ �f�t�� (5) £ �U�t � a� f�t�� � e�as £ �f�t � a�� (6) e�as £ �f�t�� � £ �U�t � a� f�t � a�� (7) £ �t f�t�� � �F ��s� (8) £ � f t � � � s �� F�s��s ----------------------------------------------------------------------- - - - - - - - - - - - - - - - - - - - - - - - - - - Untitled-1 3