# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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PROBLEMS

In each of Problems 1 through 18 find all singular points of the given equation and determine whether each one is regular or irregular.

1.

3.

5.

6.

7.

8. 9.

10.

11.

13.

15.

17.

xy

x 2(1 x )y (1 x 2)V x 2 y

+ (1 x )y + xy = 0 2.

+ (x 2)y' 3xy = 0 4.

+ x (1 x )y' + (1 + x ) y = 0 + xy' + (x2 v2)y = 0, Bessel equation (x + 3)yw 2xy' + (1 x 2)y = 0 x (1 x 2)3yw + (1 x 2)2y ' + 2(1 + x )y = 0 (x + 2)2(x 1) y" + 3(x 1)y' 2(x + 2) y = 0 x (3 x )y" + (x + 1)y' 2y = 0 (x2 + x 2)yw + (x + 1) y' + 2y = 0 12.

y" + (ln |x \)y'+ 3xy = 0 14.

x 2yw 3(sin x )y' + (1 + x 2)y = 0 16.

(sin x ) y" + xy + 4y = 0 18.

x 2(1 x )2 y'

x 2(1 x 2)y'

+ 2xy + 4y = 0 + (2/x ) y + 4y = 0

+ exy' + (3 cosx )y = 0 ; + 2(ex 1)y; + (ex cosx)y = 0 xy" + y + (cot x )y = 0 (x sin x )y" + 3y' + xy = 0

xy

x2y

In each of Problems 19 and 20 show that the point x = 0 is a regular singular point. In each

TO

problem try to find solutions of the form ^ anxn. Show that there is only one nonzero solution

n=0

of this form in Problem 19 and that there are no nonzero solutions of this form in Problem 20. Thus in neither case can the general solution be found in this manner. This is typical of equations with singular points.

19. 2xy" + 3y; + xy = 0

20. 2x2yw + 3xy; (1 + x)y = 0

260

Chapter 5. Series Solutions ofSecond Order Linear Equations

21. Singularities at Infinity. The definitions of an ordinary point and a regular singular point given in the preceding sections apply only if the point x0 is finite. In more advanced work in differential equations it is often necessary to discuss the point at infinity. This is done by making the change of variable f = 1/x and studying the resulting equation at f = 0. Show that for the differential equation P(x)y" + Q(x)y' + R(x)y = 0 the point at infinity is an ordinary point if

have Taylor series expansions about f = 0. Show also that the point at infinity is a regular singular point if at least one of the above functions does not have a Taylor series expansion, but both

do have such expansions.

In each of Problems 22 through 27 use the results of Problem 21 to determine whether the point at infinity is an ordinary point, a regular singular point, or an irregular singular point of the given differential equation.

22. y" + y = 0 23. x2y" + xy' 4y = 0

24. (1 x2)y" 2xy' + a(a + 1)y = 0, Legendre equation

25. x2y" + xy' + (x2 v2)y = 0, Bessel equation

26. y" 2xy' + Xy = 0, Hermite equation

27. y" xy = 0, Airy equation

The simplest example of a differential equation having a regular singular point is the Euler equation, or the equidimensional equation,

where a and 3 are real constants. It is easy to show that x = 0 is a regular singular point of Eq. (1). Because the solution of the Euler equation is typical of the solutions of all differential equations with a regular singular point, it is worthwhile considering this equation in detail before discussing the more general problem.

In any interval not including the origin, Eq. (1) has a general solution of the form y = c1 y1 (x) + c2y2(x), where y1 and y2 are linearly independent. For convenience we first consider the interval x > 0, extending our results later to the interval x < 0. First, we note that (xr)' = rxr1 and (xr)" = r (r 1)xr2. Hence, if we assume that Eq. (1) has a solution of the form

1 2 P (1/?) Q(1?)-

pam $ $2 .

$ [2 p (1/$) Q(1/$)

P (1/$n $ $2

5.5 Euler Equations

L [y] = x2 y" + a xy + ?y = 0,

(1)

r

y = x ,

(2)

then we obtain

L [xr ] = x 2(xr )" + ax (xr ); + ? xr = xr [r (r 1) + ar + ? ].

(3)

5.5 Euler Equations

261

If r is a root of the quadratic equation

F (r) = r (r 1) + ar + ft = 0, (4)

then L [xr] is zero, and y = xr is a solution of Eq. (1). The roots of Eq. (4) are

(a 1) ħ V(a 1)2 4ft ^

r1, r2 =--------------------2---------------- (5)

and F(r) = (r rft)(r r2). As for second order linear equations with constant coefficients, it is necessary to consider separately the cases in which the roots are real and different, real but equal, and complex conjugates. Indeed, the entire discussion in this section is similar to the treatment of second order linear equations with constant coefficients in Chapter 3 with erx replaced by xr; see also Problem 23.

Real, Distinct Roots. If F(r) = 0 has real roots rx and r2, with rx = r2, then y1 (x) = xri and y2(x) = xr2 are solutions of Eq. (1). Since W{x^, xr2) = (r2 r1)xri+''21 is nonvanishing for rx = r2 and x > 0, it follows that the general solution of Eq. (1) is

y = c1 xr1 + c2xr2 x > 0. (6)

Note that if r is not a rational number, then xr is defined by xr = er ln x.

Solve

2x2y" + 3xy' y = 0, x > 0. (7)

Substituting y = xr in Eq. (7) gives

xr [2r (r 1) + 3r 1] = xr (2r2 + r 1) = xr (2r 1)(r + 1) = 0.

Hence rj = 2 and r2 = 1, so the general solution of Eq. (7) is

y = c1 x1/2 + c2x1, x > 0. (8)

Equal Roots. If the roots rx and r2 are equal, then we obtain only one solution yj(x) = xr1 of the assumed form. A second solution can be obtained by the method of reduction of order, but for the purpose of our future discussion we consider an alternative method. Since rl = r2, it follows that F(r) = (r rl)2. Thus in this case not only does F(rj) = 0 but also F'(rl) = 0. This suggests differentiating Eq. (3) with respect to r and then setting r equal to rv Differentiating Eq. (3) with respect to r gives

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