It was stated in the Closure thatany given Cauchy - Euler equation can be reduced to a constant - coefficient equation by the change of variables x= et. In this exercise we ask you to try that idea for some specific cases; in the next exercise we ask for a general proof of the italicized claim. Lety(x(t))= Y(t), and lety′andY′denotedy/dxanddY/dt,respectively.
Show that the change of variablesx = etreduces the Cauchy - Euler equationx2y″− xy′− 3y = 0 to the constant - coefficient equationY″−2Y′−3Y = 0. Thus, show that
Y(t) = Ae−l+Be3t.Since t = In x , show thaty(x) = Ax −1+Bx3, for 4 x2y″ − y= 0.
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