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# wksht01 - MATH 152 L1 L2 Applied Linear Algebra and...

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MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Spring 2004-05 ¥ Week 01 Worksheet (January 31, 2005) : First-order differential equations Q1. (Linear vs. nonlinear) Indicate whether the following equation is linear or nonlinear . 1. dy dt + 3 t 2 y = 0 linear . 7. dy dt = t + sin y . 2. y 0 + y 2 = t . 8. dy dt 2 = y . 3. dy dt = 2 ty 2 t 2 - y 2 . 9. y 0 + y = sin - 1 t 2 , | t | < 1 . 4. dy dt + 2 y = e 2 t . 10. dy dt = 2 t - y y - t . 5. y 0 = t p 1 + y 2 . 11. dy dt = te y + t . 6. y dy dt + ty = t . 12. dx dt + 2 t + 1 2 t x = 2 x + 1 2 t . Q2. (i) (Direction field) Match the graph of the direction field to the differential equation and give a brief justification for your choice. –3 –2 –1 0 1 2 3 y(t) –3 –2 –1 1 2 3 t (a) dy dt = - ty . (b) dy dt = ty . (c) dy dt = t + y . (d) dy dt = t - y . (e) dy dt = - t + y . (ii) (Integral curve) In the above direction field graph, draw an approximate integral curve for the solution which satisfies the initial condition y (2) = 0.

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Q3. (Integrating factor) Consider the differential equation ty 0 + y = g ( t ) , t > 0 , where g ( t ) is a continuous function for t > 0. (a) Find the general solution by the method of integrating factor. (b) Show that if lim t →∞ R t 0 g ( s ) ds t = 0, then every solution of the differential equation satisfies lim t →∞ y ( t ) = 0
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wksht01 - MATH 152 L1 L2 Applied Linear Algebra and...

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