Download MA3220 (2013/14) Tutorial 5

MA3220 (2013/14)
Tutorial 5
(1) Solve the following Euler-Cauchy equation:
(a) x2 y 00 − 3xy 0 + 4y = 0
[y = (c1 + c2 log |x|)x2 ]
(b) x2 y 00 − 2y = 3x2 , y(1) = 4, y 0 (1) = 0
[x2 + 3/x + x2 log x]
(c) x2 y 00 + 7xy 0 + 13y = 0
[yx3 = c1 cos(2 log |x|) + c2 sin(2 log |x|)]
(d) xy 00 − y 0 = (3 + x)x2 ex
[c1 + c2 x2 + x2 ex ]
(e) x2 y 00 + 7xy 0 − 7y = 0, x > 0.
c2
]
x7
−1/3
[y = c1 x +
(f) 9x2 y 00 + 15xy 0 + y = 0, x > 0.
[y = (c1 + c2 ln x)x
]
(g) x2 y 00 + xy 0 + 9y = 0, x < 0.
[y = c1 cos(3 ln |x|) + c2 sin(3 ln |x|)]
(h) x3 y 000 + 4x2 y 00 − 5xy 0 − 15y = 0, x > 0.
[y = c1 x3 +
00
(i) y −
4 0
y
x
+ x62 y
2 00
2
1
(c
x2 2
cos ln x + c3 sin ln x)]
[(c1 − sin x)x2 + c2 x3 ]
= x sin x
(j) x3 y 000 − 3x y + 6xy 0 − 6y = 2x4 , x > 0; y(1) = 0, y 0 (1) = 1, y 00 (1) = −2.
[y = − 10
x + 6x2 − 3x3 + 13 x4 ]
3
(2) Find the general solution of the following (using method of reduction of order):
2 0
2y
(a) y 00 −
y + 2
= 0, y1 (t) = t + 1
t+1
t + 2t + 1
2
(b) y 00 − 4ty 0 + (4t2 − 2)y = 0, y1 (t) = et
(c) (1 + t2 )y 00 − 2ty 0 + 2y = 0, y1 (t) = t.
(3) Solve the following equations
(a) x2 y 00 + 2xy 0 + y/x2 = 0, (Hint: t = 1/x)
√
(b) x2 y 00 + xy 0 + (4x2 − 14 )y = 0. (Hint: y = z/ x)
(4) Let A be a 2 × 2 matrix. Assume that λ0 is an eigenvalue of A with multiplicity
2, and A has only one linearly independent eigenvector associated with λ0 . Let K
¯
be such an eigenvector, and let Q be a vector satisfying (A − λ0 I)Q = K. Show
¯ ¯
¯
¯
that, the general solution of ẋ = Ax is given by
x = c1 eλ0 t K + c2 eλ0 t (tK + Q).
¯
¯ ¯
(5) Let x1 , x2 , · · · , xn be solutions of ẋ = A(t)x.
If A(t) is an n × n continu-
ous matrix valued function on (a, b), show that either w(x1 , · · · , xn )(t) ≥ 0 or
w(x1 , · · · , xn )(t) < 0 on (a, b).
2
(6) Solve the following system of ODEs (i) directly, and (ii) by converting them into
2nd orderlinear ODE

3 0
x
(a) ẋ = 
2 1

(b) ẋ = 

(c) ẋ = 

(d) ẋ = 
0

1
1 −1
1 2
2 1
0

x + 

0


3
x
1


0

x + 

(convert to 2nd order ODE only)
−1/t2 1/t
1/t2
(7) Let y1 , y2 be solution of a differential equation: y 00 + p(x)y 0 + q(x)y = 0. If φ(x) is
the Wronskian of y1 and y2 , show that the following equation holds:
φ0 (x) + p(x)φ(x) = 0.
Deduce the Abel’s formula: y1 (x)y20 (x) − y2 (x)y10 (x) = ce−
R
p(x)dx
.
(8) Assume that p(x) and q(x) are continuous functions, and y1 (x) and y2 (x) are the
solutions of y 00 + p(x)y 0 + q(x)y = 0 on the interval α < x < β.
(i) Show that if y1 and y2 have maxima or minima at the same point, then they
are linearly dependent on (α, β).
(ii) Show that, if y1 and y2 are linearly independent solutions, then they cannot
have a common point of inflection in (α, β) unless p and q vanish simultaneously
there.
To discuss 1a,c,e,g,i ,2a,c, 3a, 4,5,7,8, 6a,b