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72 CHAPTER 2 Linear Second-Order Equations

16. Carry out the program of Problem 15 for the critically 17. Carry out the program of Problem 15 for the under-
damped, forced system governed by damped, forced system governed by

y + 4y + 4y = 4cos(3t). y + y + 3y = 4cos(3t).

2.5 Euler’s Differential Equation

If A and B are constants, the second-order differential equation
2

x y + Axy + By = 0 (2.15)

is called Euler’s equation. Euler’s equation is defined on the half-lines x >0 and x <0. We
will find solutions on x > 0, and a simple adjustment will yield solutions on x < 0.

A change of variables will convert Euler’s equation to a constant coefficient linear second-
order homogeneous equation, which we can always solve. Let

x = e t
t
or, equivalently, t = ln(x). If we substitute x = e into y(x), we obtain a function of t as
t
Y(t) = y(e ).
To convert Euler’s equation to an equation in t, we need to convert derivatives of y(x) to
derivatives of Y(t). First, by the chain rule, we have

d d

y (x) = (y(x)) = (Y(t))
dx dx
dY dt 1

= = Y (t).
dt dx x
Next,
d

y (x) = (y (x))
dx
d 1

= Y (t)
dx x
1 1 d

=− Y (t) + (Y (t))
x 2 x dx
1 1 dY (t) dt

=− Y (t) +
x 2 x dt dx
1 1 1

=− Y (t) + Y (t)
x 2 x x
1

= (Y (t) − Y (t)).
x 2
Therefore,
2

x y (x) = Y (t) − Y (t).

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October 14, 2010 14:12 THM/NEIL Page-72 27410_02_ch02_p43-76