Download Problem Sheet No. 8

Karlsruhe, 5th June 2014
Karlsruhe Institute of Technology
Institute for Algebra und Geometry
Prof. M. Axenovich Ph.D.
Dipl.-Math., Dipl.-Inf. J. Rollin
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Exercise Sheet No. 8
Advanced Mathematics II
Exercise 36: Find the a real-valued general solution of the following linear homogeneous differential equations
with constant coefficients
(a) y 000 (x) − 3y 00 (x) − y 0 (x) + 3y(x) = 0, x ∈ R ,
(b) y 000 (x) + 7y 00 (x) + 19y 0 (x) + 13y(x) = 0, x ∈ R ,
(c) y (4) (x) − 7y 000 (x) + 18y 00 (x) − 20y 0 (x) + 8y(x) = 0, x ∈ R.
Hint: 2 is a root of the characteristic polynomial.
Exercise 37: Find a real-valued general solution of the following differential equations and solve the initial
value problems. Give sketches of the solutions.
(a) y 00 (x) + 13y(x) = 6y 0 (x), x ∈ R, y(π) = 1, y 0 (0) = 0,
(b) 2y 00 (x) − 3y 0 (x) = 2y(x), x ∈ R, y(ln(4)) = 1, y 0 (2) = − 1e ,
(c) πy(x) − 2y 0 (x) =
−1 00
π y (x),
0
x ∈ R, y( ln(2)
π ) = 2, y (0) = π.
Exercise 38: Find a real-valued general solution of the following differential equation
u000 (x) −
5
2 00
5
u (x) + 2 u0 (x) − 3 u(x) = 0,
x
x
x
x > 0.
Exercise 39: Show that y(x) = x solves the following differential equation
(1 + x2 )y 00 (x) − 2xy 0 (x) + 2y(x) = 0 , x ∈ R.
Find another solution of the differential equation using the method of reduction of the order.
Hint: − x32+x =
2x
x2 +1
− x2 .
Exercise 40: A mass M of 5 kg stretches a spring about 0.1 m. This system is placed in a viscous fluid.
Due to the fluid a braking force of 2 N acts on the mass if the velocity equals 0.04 m/s. For the acceleration of
gravity we may assume g = 10 m/s2 .
(a) Calculate the spring constant D and the damping constant σ.
(b) Let u(t) denote the displacement of the mass from its resting position after time t. Then the
spring force is given by FF (t) = −Du(t), the damping is given by FD (t) = −σu0 (t) and the
inertia equals FT (t) = −M u00 (t). Set up a differential equation for the displacement u(t)
from the balance of forces for spring force, damping and inertia. Find the general (real)
solution.
11111111
00000000
00000000
11111111
m
00000000
(c) The mass is released 1 m from its position of rest. Calculate the solution of this initial value 11111111
00000000
11111111
00000000
11111111
problem.
F
00000000
11111111
00000000
11111111
00000000
11111111
00000000
Hint: Choose the origin at the rest position of the mass. The force of gravity is then compensated 11111111
00000000
11111111
by a part of the spring force. Therefore it does not occur in our balance of forces.
Due date: Your written solutions are due on Tuesday, 17th June 2014. Please put them into the box in the
student office until 2:00 PM.
Tutorial No. 8
Advanced Mathematics II
Exercise T22: Find a real-valued general solution of the following differential equations and solve the initial
value problem, if provided
(a) y 000 (x) − 6y 00 (x) + 12y 0 (x) − 8y(x) = 0, x ∈ R.
Hint: 2 is a root of the characteristic polynomial.
(b) y 00 (x) + y 0 (x) − 2y(x) = 0, x ∈ R,
(c) 9y(x) + 12y 0 (x) = −4y 00 (x), x ∈ R, y(0) = 4, y 0 (0) = 0,
π
(d) y 00 (x) − 2y 0 (x) + 5y(x) = 0, x ∈ R, y(π) = 0, y 0 ( π4 ) = e 4 .
Exercise T23: Find a general solution of the following differential equation
x3 y 000 (x) − 3x2 y 00 (x) + 7xy 0 (x) − 8y(x) = 0 .
2
Exercise T24: Show that u(x) = ex is a solution of the following homogeneous differential equation
u00 (x) − 2xu0 (x) − 2u(x) = 0,
x ∈ (0, ∞) .
2
Find another solution using the method of reduction of the order, that is not a constant multiple of ex .
R
2
2
Hint: There is no explicit form for the antiderivative e−x dx of e−x . So just keep the integral in the answer.
For detailed information regarding this course visit the following web page:
www.math.kit.edu/iag6/lehre/am22014s/en
Tutorial:
Friday, 13th June 2014, 9:45 AM