# MIT 18 034 - LECTURE NOTES (2 pages)

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## LECTURE NOTES

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## LECTURE NOTES

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Pages:
2
School:
Massachusetts Institute of Technology
Course:
18 034 - Honors Differential Equations
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MIT OpenCourseWare http ocw mit edu 18 034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 034 Practice Midterm 3 Notation d dt 1 a If f E and F s L f t show that lims F s 0 s 1 b Find the inverse Laplace transform of F s log s 1 2 a Sketch the graph of f t 1 5 h t 5 t 5 h t 10 t 10 where h t is the unit step function or the Heaviside function b Find the solution of the initial value problem y 4y f t y 0 0 y 0 0 Sketch the graph of the solution c Compute the left and the right limits of y t at t 5 and t 10 3 Consider two vectors y1 t t 1 and y2 t t2 2t a In which intervals are y1 and y2 are linearly independent b Find a system of differential equations satis ed by y1 and y2 4 Find the general solution of x 1 1 x y 1 3 y 5 Let A 0 1 1 0 a Show that A2 I b Show that e c Find the general solution of At cos t sin t sin t cos t x x A y y d Sketch solutions in the x y plane and discuss their behavior 1

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