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EGR/MA 265, Math Tools for Engineering Problem Solving February 08, 2010, 50 minutes
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TEST I
Problem 1 (16P) Determine the order of the following ODEs.(4P+4P+4P+4P) Also, state if they are linear or non-linear.
(a) y′^ + sin(y) = x (b) y(6)^ − y(3)^ = cos(xy) (c) yy′′^ = ex (d) y^ − x^ sin( (^2) y′ x)= ex
Problem 2 (12P) (a) Which of the following functions are solutions of x^4 y′^ + 2xy^2 = 4x^5? (8P) y 1 = −x^2 , y 2 = x, y 3 = x^2 , y 4 = − 2 x^2.
(b) Which of the functions from part (a) solve the initial value problem y(0) = 0? (4P) x^4 y′^ + 2xy^2 = 4x^5 ,
(c)Uniqueness Theorem for first order ODEs? If yes, why? If no, why not? (5P∗^ (Bonus) Does your answer to part (b) agree with the content of the Existence and∗)
Problem 3 (12P) (a) In the 3direction field for × 3-grid of points y′ (^) = x (^2) (y − 2). (8P)x = 0, 1 , 2 and y = 0, 1 , 2 provided in the figure below draw a
(b) Without solving the DE, use the direction field to read off the solution of the IVP y′ (^) = x (^2) (y − 2), y(1) = 2. (4P)
Problem 6 (12P) Solve the IVP y′ (^) − y (^2) sin(x) = 0, y ( (^) π 3
Problem 7 (12P) The mass of a radioactive material is given byand the mass in grams. An initial mass of m m(0) = 100 grams decays at a constant rate(t), where the time t is measured in years k = m′(t)/m(t). After 1 year 80 grams of the material are left. (Note: Your answers will contain natural logarithms which do not need to be evaluated.)
(a) Find the decay rate k by solving the differential equation for m(t). (8P)
(b) Find an expression for the time is the so-called half-life of the material). (4P) th at which only 50 grams of the material are left (th
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