practicefinalODE2009

# practicefinalODE2009 - y 00 4 y = 0 using the power series...

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Practice Final Exam Professor Cristian Virdol E 1210 Section 001 December 6, 2009 Name: To receive full credit, you must explain your answers. 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 100 1

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1. Solve the diﬀerential equations: (a) y 2 dx + (2 xy - y 2 e y ) dy = 0 (b) cos ty 0 - sin ty = e - 3 t 2
2. Find the general solution of the equation: y 00 + 3 y 0 - 4 y = t 2 + t cos t + e 4 t 3

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3. Determine (without solving the equation) the largest interval in which the solution of the initial value problem is certain to exist: ± t 2 ( t - 8) y 0 + 1 t - 7 y = e 2 t y (3) = 1 Justify your answer. 4
4. Consider the equation y 0 = e y ( y 2 - 5 y + 4). Find the equilibrium points, draw the phase line and classify the equilibrium points as stable, or un- stable. 5

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5. Find the general solution of the system x 0 = ± 1 1 4 - 2 ² x + ± - 2 e t e - 3 t ² 6
6. Find the general solution of the equation

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Unformatted text preview: y 00 + 4 y = 0 using the power series method. 7 7. Find the solution of the initial value problem y 00 + 4 y = δ ( t-2 π ) , y (0) = 1 , y (0) = 1 8 8. Find the Laplace transform of each of the following functions: (a) t 2 sin 3 t + 3 t 2 + 2 t (b) δ ( t-5) sin 3 t 9 9. Solve the diﬀerential equation by using the Laplace transform y ( t ) + y ( t ) = Z t sin( t-ξ ) y ( ξ ) dξ, y (0) = 1 10 10. For the system x = ± 3-4 1-1 ² x classify the type of the critical point (0 , 0) and determine whether it is stable, asymptotically stable or unstable. 11...
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practicefinalODE2009 - y 00 4 y = 0 using the power series...

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