By F. Marotto

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But P(r fc+1 ) = Afc+1. 7. Let w(x) and Ì;(X) be functions that possess two continuous derivatives on an interval / and which are such that W(x; u, v) Φ 0 for x in /. Show that the equation ' y y' y" u(x) u\x) u"{x) = 0 v(x) v'(x) v"(x) is a linear homogeneous second-order differential equation for which u(x) and v(x) are solutions on /. 8. By using the result of Problem 7, construct a linear homogeneous secondorder differential equation that has the given functions as solutions on the given intervals.

F(x). 85) by e~r2X, differentiate M2 + 1 times, and then multiply through by eriX. , s. 86) where fs(x) is a polynomial of the same degree as ps(x), namely, Ms. 86) by e~rsX and differentiating Ms 38 I Linear Differential Equations times, we find that dxM' fs(x) = 0. But this is impossible, since fs(x) is of degree Ms. Hence our assumption that the n solutions were linearly dependent is false; they must be linearly independent. 88) the resulting set of n real solutions is still linearly independent.

Use the result of Problem 5 to find particular solutions of the equations in Problems 1(c) and 1(d). 7. If the function yp(x) is a solution of the equation P(D)y = Aeiax (A is real), then the real and imaginary parts of yp(x) are real solutions of the equations P(D)y = A cos ax, P(D)y = A sin ax, respectively. Use this fact to find particular solutions of the equations in Problems l(i) and l(j). 14 Applications In this section we shall consider some elementary problems in mechanics and in electric circuit theory that lead to initial value problems for linear differential equations.