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Unformatted text preview: MATH 24-Exam 1 Spring 2010 Solution (1) For IVP y = y (3 t- 1) , y (0) = 2 find Euler approximate solution at t=1 using step size h = 0 . 5 . Do you see any problems with this approximation? Eulers formula is y n +1 = y n + hf ( t n ,y n ) and t n +1 = t n + h t y y 2- 2 . 5 1 2 1 2 y (1) 2 although this approximation is incorrect because y is undefined at t = 1 3 and with the step size of .5 were stepping right over the singularity. (2) Match each of the following differential equations with one of the direction fields. (3pts each) ( i ) d y d t = y ( y 2- 2 ) (B) ( ii ) d y d t = t (2- y ) (A) Equilibrium point at y = 2 as opposed to the isocline at t = 2 in (iii) ( iii ) d y d t = y (2- t ) (D) ( iv ) d y d t = sin y (C) Unstable equilibrium at y = 0 as opposed to stable at y = 0 in (i) (A) (B) (C) (D) MATH 24-Exam 1 Spring 2010 (3) Consider IVP ty- 3 y = t 4 , y (1) = 4 a. What does the Picards Theorem tell you for this problem?...
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