If the functions y1 and y2 are a fundamental set of

Chapter 3, Problem 33

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If the functions y1 and y2 are a fundamental set of solutions of y+ p(t)y+ q(t)y = 0,show that between consecutive zeros of y1 there is one and only one zero of y2. Notethat this result is illustrated by the solutions y1(t) = cost and y2(t) = sin t of the equationy+ y = 0.Hint: Suppose that t1 and t2 are two zeros of y1 between which there are no zeros of y2.Apply Rolles theorem to y1/y2 to reach a contradiction.Change of Variables. Sometimes a differential equation with variable coefficients,y+ p(t)y+ q(t)y = 0, (i)can be put in a more suitable form for finding a solution by making a change of the independentvariable. We explore these ideas in 34 through 46. In particular, in weshow that a class of equations known as Euler equations can be transformed into equationswith constant coefficients by a simple change of the independent variable. 35 through 42 are examples of this type of equation. determines conditions under which themore general Eq. (i) can be transformed into a differential equation with constant coefficients. 44 through 46 give specific applications of this procedure.