24.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. Note that this result is illustrated by the solutions y1(t) = cos t and y2(t) = sin t of the equation y″ + y = 0. Hint: Suppose that t1 and t2 are two zeros of y1 between which there are no zeros of y2. Apply Rolle's theorem to y1/ y2 to reach a contradiction. Change of Variables. Sometimes a differential equation with variable coefficients, (32) y′′+p(t)y′+q(t)y=0 can be put in a more suitable form for finding a solution by making a change of the independent variable. We explore these ideas in Problems 25 through 36. In particular, in Problem 25 we show that a class of equations known as Euler equations can be transformed into equations with constant coefficients by a simple change of the independent variable. Problems 26 through 31 are examples of this type of equation. Problem 32 determines conditions under which the more general equation (32) can be transformed into a differential equation with constant coefficients. Problems 33 through 36 give specific applications of this procedure.
24.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. Note that this result is illustrated by the solutions y1(t) = cos t and y2(t) = sin t of the equation y″ + y = 0.
Hint: Suppose that t1 and t2 are two zeros of y1 between which there are no zeros of y2. Apply Rolle's theorem to y1/ y2 to reach a contradiction.
Change of Variables. Sometimes a
can be put in a more suitable form for finding a solution by making a change of the independent variable. We explore these ideas in Problems 25 through 36. In particular, in Problem 25 we show that a class of equations known as Euler equations can be transformed into equations with constant coefficients by a simple change of the independent variable. Problems 26 through 31 are examples of this type of equation. Problem 32 determines conditions under which the more general equation (32) can be transformed into a differential equation with constant coefficients. Problems 33 through 36 give specific applications of this procedure.
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