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practiceMidterm1ODE1

# practiceMidterm1ODE1 - (a Determine(without solving the...

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Practice Midterm I Professor Cristian Virdol E 1210 Section 001 September 23, 2009 Name: To receive full credit, you must explain your answers. 1 16 2 36 3 16 4 16 5 16 Total 100 1

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1. (16 points) (a) Solve the differential equation: y 0 + 3 y = 4 t (b) Solve the initial value problem: sin t · y 0 + cos t · y = sin t y ( π 2 ) = 1 2
2. (36 points) Solve the differential equations: (a) (3 t + 1) y 0 = y ( y + 1) (b) y 0 = 1+ x 2 xy + xy 4 3

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(c) 3 x 2 y 2 + (2 x 3 y + 2 y ) y 0 = 0 (d) ( x 2 y 2 + xy ) + ( x 3 y + x 2 ) y 0 = 0 knowing that it becomes exact when multiplied by an integrating factor μ ( x ) depending only on x . 4
3. (16 points) Consider the equation y 0 = e y ( y 2 - 7 y + 12) (a) Find the equilibrium points, draw the phase line and classify the equilibrium points as stable or unstable. (b) Draw the solution for y (0) = 7 2 and describe its concavity 5

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4. (16 points)
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Unformatted text preview: (a) Determine (without solving the equation ) the largest interval in which the solution of the initial value problem is certain to exist: ‰ y + 1 ln( t-2) y = t-4 t-5 y (3) = 1 Justify your answer. (b) Consider the initial value problem: ‰ ( t 2 + y 2 + 1) y = t + y y ( t ) = y For what values of ( t ,y ) we are guaranteed to have a unique solution in some interval around t ? Justify your answer. 6 5. (16 points) (a) Solve the initial value problem ‰ y 00 + y-20 y = 0 y (0) = 1 ,y (0) = 1 (b) Find a diﬀerential equation whose general solution is c 1 e t 4 + c 2 e-t 5 . 7...
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practiceMidterm1ODE1 - (a Determine(without solving the...

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