## Transcribed Text

1. Show that J1;2(x) = (:
x
)
1
1
2
sinx and that L1;2(x) = (,,r2
x
) 1
1
2
cosx
( 2 ) 1
1
2 { sin x
} Hence, show that J3;2 ( x) =
1rx x- cos x
2. Show that the Bessel equation of order one-half,
x
2
y" + xy' + ( x
2 - ) y = 0
can be reduced to the equation
V
11 + V = 0
by the change of dependent variable y = x-1
1
2
v(x).
From this conclude that
Y1(x) = x-1
/
2 cosx and Y2(x) = x-1
!
2 sinx
are solutions of the Bessel equation of order one-half.
3. Show that Jo'(x) = -J1(x)
4. Prove the following results.
(a)
(c)
1 £ {Jo(x)} =
yp2 + 1
and that
(b)
(d)
5. Evaluate the integral fo,i;
Jo(x - t)Jo(t) dt.
1
Ji'(x) = Jo(x) - _!_J1(x)
X
1 £ {Io(x)} =
yp2 -1
6. (a) Show that Jo (x) = - cos(xsin0) d0.
7T" 0 11,r
211 cosxt (b) Hence, show that J0 (x) = - vT=t2" dt
7T" 0 1 - t2
7. Use the generating function to show that
Jn (X + y) = I: Jr(x)Jn-r(Y)
r=-oo
dF(p) 8. (a) Prove the result .C {xf(x)} = -
(b) Use the result in (a) to transform the differential equation
d
2
y dy x
dx2 + dx
+
xy = O
into another of the first order, and solve it. Then invert to show that the solution of
the given differential equation when y(O
) = 1 is Jo (x).
9. Show that the solution of the differential equation
is y = AJn(kx) + BYn (kx)
10. Consider the differential equation
x
2
y" + cxy' + (x2 - n2
) y = 0
Let 1- 2r = c.
Make the substitution y = x
rY and obtain the Bessel differential equation.
Hence solve the differential equations
(a) x
2 y" + 2xy' + 4x
2
y = 0
2
(b) xy" - 3y' + xy = 0
11. Solve the differential equation
x
2
y" - 2xy' + 2 ( x2 + 1) y = 0
12. Solve the following differential equations.
(a) xy" - y' + 4x3y = 0 (try x
2 = X)
(b) x
2 y" + (2x
2 + x) y' + (2x
2 + x) y = 0
(c) y" + x
2
y = 0 (try x
2 = X)
( d) xy" + ( 1 + 4x2
) y' + x ( 5 + 4x2
) y = 0
13. ( a) By a suitable change of variables it is often possible to transform a differential equation
with variable coefficients into a Bessel equation of a certain order.
For example, show that a solution of
x
2
y" + ( a2/32x
2(3 + i- v
2/32
) y = 0 x > 0
is given by y = x
1l2 f ( ax
f3
) where f (0 is a solution of the Bessel equation of order
V.
(b) Using the result of Question 14( a), show that the general solution of the Airy equation
is
y" + xy = 0 X > 0
y = x
112 [cif1 (x
312
) + c2f2 (x
312)]
where fi(O and h() are linearly independent solutions of the Bessel equation of
order one-third.
3
14. (a) Show that a solution of
x>O
is given by y = x
0 f (f3x7), where f (c;) is a solution of the Bessel equation of order n.
(b) Using the result of Question 14(a), solve the following differential equations.
(i) x
2y" + xy' + 4 (x4 - n
2
) y = 0
(ii) xy" + y = 0
(iii) x
1f2y
11 + y = 0
15. It can be shown that Jo (x) has infinitely many zeros for x > 0. In particular the first three
zeros are approximately 2.405, 5.520, and 8.653.
Letting Aj
,j = 1,2, ... denote the zeros of Jo (x), it follows that
1, x= 0,
0 X = l.
Verify that y = Jo (AjX) satisfies the differential equation
Hence show that
I/ 1 / \ 2 y + -y + /\j y = ' 0
X
X > 0.
if
This important property of Jo (>,.j
x), known as the orthogonality property, is useful in
solving boundary value problems.
4
Hint: Write down the differential equation for J0 ( AiX).
Multiply it by xJo (.Xj
x) and subtract it from xJo (.Xix) times the differential equation for
Jo (.Xj
x)
Then integrate from O to 1.

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