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Unformatted text preview: Department of Mathematical Sciences University of Delaware Prof. T. Angell September 16, 2009 Mathematics 351
Exercise Sheet 2 Exercise 6: Consider the initial value problem for the nonhomogeneous linear equation dy = −y + x , y (0) = 4 . dx (a) Show that y (x) = x − 1 is a solution of the diﬀerential equation. (b) Find a one parameter family of solutions of the diﬀerential equation. Determine the parameter value corresponding to the given initial condition. (c) Show that the graphs of all solutions of the diﬀerential equation are asymptotic to the line y = x − 1 as x → ∞. Exercise 7: Find all solutions of the diﬀerential equation t dx − 4 x = t6 et , t > 0 . dt Exercise 8: Show that the functions y1 (t) ≡ 0, and y2 (t) = (t − to )3 are both solutions of the initial value problem y = 3 y 2/3 , y (to ) = 0 on −∞ < t < ∞. Graph both solutions for the successive values of to = −1, 0, 1. Exercise 9: An equation of the form y + p(t)y = q (t)y n , is called a Bernoulli equation. (a) Solve Bernoulli’s equation when n = 0 and when n = 1. (b) Recalling that if n = 0, 1 then the substitution v = y 1−n reduces Bernoulli’s equation to a linear equation, solve t2 y + 2 t y − y 3 = 0. (c) The equation y = y − σ y 3 with > 0 and σ > 0 occurs in the study of the stability of ﬂuid ﬂow. Solve the equation. See pp. 68-69 of your text. Exercise 10: The equation dy = q1 (t) + q2 (t) y + q3 (t) y 2 , dt is known as a Riccati equation. Such equations are important in feedback control theory. Suppose that some particular solution y1 of this equation is known. A more general solution containing one arbitrary constant can be obtained through the substitution y = y1 (t) + 1 . v (t) (a) Show that, if y is to be a solution of the Riccati equation, then v (t) must satisfy the ﬁrst order linear equation dv = − (q2 + 2 q3 y1 ) v − q3 . dt (NOTE: v will contain an arbitrary constant.) (b) Given that a particular solution of y = 1 + t2 − 2t y + y 2 is y1 (t) = t, solve the Riccati equation. See also Problem #35 on p. 70 of your text. Due Date: Wednesday, Sept. 23, in class ...
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This note was uploaded on 12/02/2009 for the course MATH 352 taught by Professor Staff during the Spring '08 term at University of Delaware.
- Spring '08