Department of MathematicalSciencesUniversity of DelawareProf. T. AngellSeptember 16, 2009Mathematics 351Exercise Sheet 2Exercise 6: Consider the initial value problem for the nonhomogeneous linear equationdydx=−y+x ,y(0) = 4.(a) Show thaty(x) =x−1 is a solution of thedifferential equation.(b) Find a one parameter family of solutions of the differential equation.Determine the parametervalue corresponding to the given initial condition.(c) Show that the graphs ofallsolutions of the differential equation are asymptotic to the liney=x−1asx→ ∞.Exercise 7: Find all solutions of the differential equationtdxdt−4x=t6et, t >0.Exercise 8: Show that the functionsy1(t)≡0, andy2(t) = (t−to)3are both solutions of the initialvalue problemy= 3y2/3, y(to) = 0 on−∞< t <∞. Graph both solutions for the successive values ofto=−1,0,1.Exercise 9: An equation of the formy+p(t)y=q(t)yn, is called a Bernoulli equation.(a) Solve Bernoulli’s equation whenn= 0 and whenn= 1.(b) Recalling that ifn= 0,1 then the substitutionv=y1−nreduces Bernoulli’s equation to a linearequation, solvet2y+ 2
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Constant of integration, Prof. T. Angell, Department of Mathematical Sciences University of Delaware