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home2_09F - Department of Mathematical Sciences University...

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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 dx = y + x , y (0) = 4 . (a) Show that y ( x ) = x 1 is a solution of the di ff erential equation . (b) Find a one parameter family of solutions of the di ff erential equation. Determine the parameter value corresponding to the given initial condition. (c) Show that the graphs of all solutions of the di ff erential equation are asymptotic to the line y = x 1 as x → ∞ . Exercise 7 : Find all solutions of the di ff erential equation t dx dt 4 x = t 6 e t , t > 0 . Exercise 8 : Show that the functions y 1 ( t ) 0, and y 2 ( t ) = ( t t o ) 3 are both solutions of the initial value problem y = 3 y 2 / 3 , y ( t o ) = 0 on −∞ < t < . Graph both solutions for the successive values of t o = 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 t 2 y + 2
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