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5.5 Euler Equations
(11), and (18) to obtain real-valued solutions valid in any interval not containing the origin (also see Problems 30 and 31). These results are summarized in the following theorem.
Theorem 5.5.1 The general solution of the Euler equation (1),
x2 y" + a xy' + ft y = 0,
in any interval not containing the origin is determined by the roots equation r1 and r2 of the
F (r) = r (r — 1) + ar + ft = 0.
If the roots are real and different, then
y = cftx |T + c2|x T2. (24)
If the roots are real and equal, then
y = (c1 + c2ln |x|)|x|T. (25)
If the roots are complex, then
y = |x|A[c1 cos(iln |x|) + c2sin(i ln |x|)], (26)
where rp r2 = X ± i[i.
The solutions of an Euler equation of the form
(x — x0)2y" + a(x — x0 )y' + fty = 0 (27)
are similar to those given in Theorem 5.5.1. If one looks for solutions of the form y = (x — x0)r, then the general solution is given by either Eq. (24), (25), or (26) with x replaced by x — x0. Alternatively, we can reduce Eq. (27) to the form of Eq. (1) by making the change of independent variable t = x — x0. The situation for a general second order differential equation with a regular singular point is analogous to that for an Euler equation. We consider that problem in the next section.
PROBLEMS In each of Problems 1 through 12 determine the general solution of the given differential equation
that is valid in any interval not including the singular point.
1. x 2 y' + 4xy' + 2y = 0 2. (x + 1)2y + 3 (x + 1) y+ 0.75 y = 0
3. x 2 y' — 3xy; + 4y = 0 4. x 2 y" + 3xy' + 5y = 0
5. x 2 y — xy + y = 0 6. (x — 1)2y" + 8(x - 1) y' + 12y = 0
7. x 2 y + 6xy — y = 0 8. 2x 2 y w — 4xy; + 6y =0
9. x 2 y — 5xy + 9y =0 10. (x — 2)2y" + 5(x - 2)y' + 8y = 0
11. x 2 y + 2xy' + 4y =0 12. x 2 y" — 4xy' + 4y = 0
In each of Problems 13 through 16 find the solution of the given initial value problem. Plot the graph of the solution and describe how the solution behaves as x ^ 0.
? 13. 2x2y" + xy' — 3y = 0, y(1) = 1, y'(1) = 4
? 14. 4x2y" + 8xy'+ 17y = 0, y(1) = 2, y'(1) = -3
? 15. x2y" — 3xy; + 4y = 0, y(— 1 ) = 2, y'(—1) = 3
? 16. x2 y" + 3xy; + 5y = 0, y(1) = 1, y'(1) = — 1
Chapter 5. Series Solutions of Second Order Linear Equations
17. Find all values of a for which all solutions of x2y" + axy' + (5/2)y = 0 approach zero as x ^ 0.
18. Find all values of f for which all solutions of x2y" + fiy = 0 approach zero as x ^ 0.
19. Find y so that the solution of the initial value problem x 2y" — 2y = 0, y(1) = 1, y;(1) = Y is bounded as x ^ 0.
20. Find all values of a for which all solutions of x2 y" + a xy' + (5/2) y = 0 approach zero as x ^ ?.
21. Consider the Euler equation x 2y" + a xy' + fiy = 0. Find conditions on a and fi so that
(a) All solutions approach zero as x ^ 0.
(b) All solutions are bounded as x ^ 0.
(c) All solutions approach zero as x ^ ?.
(d) All solutions are bounded as x ^ ?.
(e) All solutions are bounded both as x ^ 0 and as x ^ ?.
22. Using the method of reduction of order, show that if r1 is a repeated root of r (r — 1) + ar + f = 0, then xr1 and xr1 ln x are solutions of x 2y" + a xy' + fy = 0 for x > 0.
23. Transformation to a Constant Coefficient Equation. The Euler equation x2y" + a xy' + fy = 0 can be reduced to an equation with constant coefficients by a change of the independent variable. Let x = ez, or z = ln x, and consider only the interval x > 0.
(a) Show that
dy 1 dy d 2y 1 d2y 1 dy
dx x dz dx2 x2 dz2 x2 dz
(b) Show that the Euler equation becomes
d2 y dy
—2 + (a — 1) -j + fy = 0-
Letting r1 and r2 denote the roots of r2 + (a — 1)r + f = 0, show that
(c) If r1 and r2 are real and different, then
y = c1er1z + c2er2z = c1 xr1 + c2xr2 .
(d) If r1 and r2 are real and equal, then
y = (C1 + C2z)er1z = (C1 + c2 ln x)xr1 .
(e) If r1 and r2 are complex conjugates, r1 = X + i p,, then
y = eXz[c1 cos(pz) + c2 sin(pz)] = xX[c1 cos(p lnx) + c2 sin(p lnx)].
In each of Problems 24 through 29 use the method of Problem 23 to solve the given equation for x > 0.
24. x2y" — 2y = 0 25. x2y" — 3xy' + 4y = lnx
26. x2yw + 7xy' + 5y = x 27. x2y" — 2xy' + 2y = 3x2 + 2 lnx
28. x2y" + xy' + 4y = sin(lnx) 29. 3x2y" + 12xy'+ 9y = 0
30. Show that if L [y ] = x 2y" + a xy' + fy, then
L [(—x ) ] = (—x )rF (r )
for all x < 0, where F (r ) = r (r — 1) + ar + f. Hence conclude that if ^ = r2 are roots
of F (r) = 0, then linearly independent solutions of L [y] = 0 for x < 0 are (—x)r1 and
( —x )r2.
31. Suppose that xr1 and xr2 are solutions of an Euler equation for x > 0, where r1 = r2, and r1 is an integer. According to Eq. (24) the general solution in any interval not containing the origin is y = c1 |x |r1 + c2|x |r2. Show that the general solution can also be written as
5.6 Series Solutions near a Regular Singular Point, Part I
y = kJxr1 + k2 |x |r2.
Hint: Show by a proper choice of constants that the expressions are identical for x > 0, and by a different choice of constants that they are identical for x < 0.