MTH 304 Final, Fall 2008

Lebanese American University Diff.Equations Fall 2008 Byblos Final Exam 2 Hours Name : ------------------------------------------------------------------------- - - - - - - - - - - - - - - - - - - - - - - - - - - - 1. Solve, by 2 different methods, the 2ndorder LDE : y� � 4 y� � 4 y � e�2 x a) using the method of variation of parameters. b) using the method of undetermined coefficients. (15 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2. Solve the 2ndorder LDE : x2 y� � 3 x y� � 5 y � 2 � x2 (10 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 3. Consider the linear system x� � x � y y� � x � 3 y 3. Consider the linear system x� � x � y y� � x � 3 y a) Find the general solution to the system. b) Determine the type and stability of the critical point O. c) Draw the phase portrait. (15 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4. Using the improved-Euler method, solve numerically the IVP y� � x�x � y� y�3� � 2 . Do 2 steps with h � 0.5 (10 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 5. Using power series, solve the LDE �x2 � x� y� � y� � y � 0. (10 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Untitled-1 1 6. You parked your car in a garage that has a constant temperature of 30 °C. Two hours later, the engine temperature was 40°C. At the time t � 3 hours, its temperature decreased to 35°C. What was the initial temperature of the engine when you parked the car. (10 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Turn Over - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - --------------------------------- Formulas: 1) £ �y�� � s £ �y� � y�0� 2) £ �y�� � s2 £ �y� � s y�0� � y��0� 3) £ �ea t f�t�� � F�s � a� 4) £ �� 0 t f�u��u� � 1 s £ �f�t�� 5) £ �U�t � a� f�t�� � e�a s £ �f�t � a�� 6) e�a s £ �f�t�� � £ �U�t � a� f�t � a�� 7) £ �t f�t�� � �F ��s� 8) £ � f�t� t � � � s �� F�s��s 9) �f � g� �t� � � 0 t f�t � u� g�u��u 10) £ �f � g� � £ �f� £ �g� Untitled-1 2 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 7. Using convolution product, solve the IVP: y� � 4 y � t y�0� � 0 , y��0� � 0 (10 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 8. Find the Laplace Transform of the function: a) f�t� � t cosh 2 t b) f�t� � U�t � 2� sin t ( 8 pts) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 9. The currents i1 and i2 in an electric circuit satisfy the system i1 � � 2 i2 � 1 i2 � � 2 i1 � �2 t where i1�0� � 1 , i2�0� � 0 Find the currents using Laplace Transforms method (12 pts) ---------------------------------------------------------------------------- - - - - - - - - - - - - - - - - - - - - - - - Untitled-1 3