2326 Solutions to Problem Sheet 21
Second Order Differential Equations I
Question 1: Find the general (real solution of each of the following equations).
(a) y 00 − 7y 0 + 10y = 0
y(x) = C1 e5x + C2 e2x
(b) y 00 + 6y 0 + 10y = 0
y(x) = e3x (C1 cos 4x + C2 sin 4x)
(c) y 00 + 25y = 0
y(x) = C1 cos 5x + C2 sin 5x
(d) y 00 − 5y 0 − 14y = 0
y(x) = C1 e−2x + C2 e7x
(e) y 00 − 4y = 0
y(x) = C1 e2x + C2 e−2x
Question 2: Find the general (real solution of each of the following equations).
(a) y 00 − 7y 0 + 10y = x3 e5x , (F)
y(x) = C2 e5x + C1 e2x +
1
x(−8 + 12x − 12x2 + 9x3 )e5x
108
(b) y 00 + 6y 0 + 10y = cos 2x
y(x) = C2 e−3x sin x + C1 e3x cos x +
1
1
1
sin 2x +
cos 2x
15
30
Esther Vergara Diaz, [email protected], see also http://www.maths.tcd.ie/~evd
1
(c) y 00 + 25y = x sin x
y(x) = C2 sin 5x + C1 cos 5x −
1
1
cos(x) + x sin x
288
24
(d) y 00 − 5y 0 − 14y = e−2x
1
y(x) = C2 e−2x + C1 e7x − xe−2x
9
(e) y 00 − 4y = 1 + x + x5 , (F)
y(x) = C2 e−2x + C1 e2x −
1 17
5
1
− x − x3 − x 5
4
8
4
4
(f) y 00 − y 0 = x2
1
y(x) = −x2 − x3 + C1 ex − 2x + C2
3
(g) y 00 − 2y 0 + y = ex + e2x , (F)
1
y(x) = C2 ex + C1 ex x + ex x2 + e2x
2
(h) y 00 + y = ex sin 2x, (F)
y(x) = C2 sin x + C1 cos x −
1 x
e (sin 2x + 2 cos 2x)
10
Question 3: Find the solution to the following initial value problems.
(a) y 00 − 7y 0 + 10y = x3 e5x ,
y(0) = 3, y 0 (0) = 0
160 5x
1
2x
2
3 5x
y(x) = 403
81 e − 81 e + 108 x(−8 + 12x − 12x + 9x )e
(b) y 00 + 6y 0 + 10y = cos 2x,
y(0) = 0,
y 0 (0) = 1, (F)
23 ∗ e−3x sin x − 1 e−3x cos x + 1 sin 2 ∗ x + 1 cos 2x
y(x) = 30
30
15
30
2
Question 4: ‘Euler’s equidimensional equation’ is:
x2 y 00 (x) + axy 0 (x) + by(x) = 0
where a, b are constants. If the independent variable is changed to z = ln x (for
x > 0) show that y(z) satisfies an equation with constant coefficients. (F)
Hence obtain the general solution of each of the following equations:
(a) x2 y 00 + 3xy 0 + 10y = 0,
(b) x2 y 00 + 5xy 0 + 4y = 0, (F)
(c) x2 y 00 + 2xy 0 − 12y = 0.
By the chain rule:
y 0 (z) = y 0 (x)x0 (z) = y 0 (x)ez = xy 0 (x),
y 00 (z) = y 00 (x)x0 (z)2 + y 0 (x)x00 (z)
= y 00 (x)e2z + y 0 (x)ez
= x2 y 00 (x) + xy 0 (x).
Therefore
x2 y 00 (x) + axy 0 (x) + by(x) = y 00 (z) + (a − 1)y 0 (z) + by(z)
and the associated constant coefficient equation is:
y 00 + (a − 1)y 0 + by = 0.
1
c1 cos(3 log x) + c2 sin(3 log x)
x
1
(b) y = 2 (c1 + c2 log x) (c) y = c1 x3 + c2 x−4 .
x
(a) y =
3
Question 1. Extra Problems: Find the general (real) solution of each of
the following equations.
(a) y 000 − 6y 00 + 11y 0 − 6y = 0, (F)
y(x) = C1 ex + C2 e3x + C3 e2x
(b) y 000 + 4y 00 + 4y 0 = 0,
y(x) = C1 + C2 e−2x + C3 e2x x
(c) y (4) + 4y 000 + 6y 00 + 4y 0 + y = 0,
y(x) = C1 e−x + C2 e−x x + C3 e−x x2 + C4 e−x x3
(d) y (4) − 16y = 0,
y(x) = C1 e−2x + C2 e2∗x + C3 sin(2x) + C4 cos(2x)
(e) y (6) + 8y (4) + 16y 00 = 0, (F)
y(x) = C1 + C2 x + C3 sin(2x) + C4 cos(2x) + C5 sin(2x)x + C6 cos(2x)x
(f) y (6) + 4y (4) + 4y 00 = 0,
√
√
√
√
y(x) = C1 +C2 x+C3 sin( 2x)+C4 cos( 2x)+C5 sin( 2x)x+C6 cos( 2x)x
(g) y (4) + 4y 000 + 8y 00 + 8y 0 + 4y = 0,
y(x) = C1 e−x sin(x) + C2 e−x cos(x) + C3 ∗ e−x sin(x)x + C4 e−x cos(x)x
(h) y (4) + 16y = 0 (F)
√
y(x) = −C1 e−
2x
√
√
√
√
√
√
√
sin( 2x)−C2 e 2x sin( 2x)+C3 e− 2x cos( 2x)+C4 e 2x cos( 2x)
Question 2. Extra Problems: Find the general (real) solution of each of
the following equations.
(a) y 000 − y 0 = ex + e−x ,
y(x) = ex C2 − C1 e−x + 21 xe−x + 12 e−x + C3
4
(b) y 000 + 3y 00 + 3y 0 + y = xe−x , (F)
1 x4 ex + C e−x + C e−x x + C e−x x2
y(x) = 24
1
2
3
(c) y 000 − y = ex .
1
y(x) = − 21 e−x + C1 ex + C2 e− 2 x cos
5
√ √ 3 x + C e−12x sin
3
3
2
2 x