Download Problem Sheet No. 10

Karlsruhe, 19th 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. 10
Advanced Mathematics II
Exercise 46: Find the roots of the characteristic polynomials of the following differential equations. Give a
candidate for a particular solution taking into consideration possible repetitions.
(a) y 00 (x) + y(x) = x sin x,
(b) y 000 (x) − 4y 00 (x) − 2y 0 (x) + 20y(x) = x2 ex ,
(c) y 000 (x) + y 00 (x) − 6y 0 (x) = xe2x + 2e−3x ,
(d) y (4) (x) + 4y 000 (x) + 6y 00 (x) + 4y 0 (x) + 5y(x) = −8 cos x − 8 sin x.
Hint: A solution of the homogeneous equation is given by y(x) = x cos(x).
Exercise 47: Solve the following initial value problem
y 000 (x) + 3y 00 (x) + 4y 0 (x) − 8y(x) = (7 − 13x)ex ,
y(0) = y 00 (0) = 2,
x ∈ R,
y 0 (0) = 0.
Exercise 48: Use the power series method to find a solution of the following (incomplete) initial value problem
xu00 (x) + 4u0 (x) + 3u(x) = 3,
u(0) = 2.
For which x ∈ R does the series converge?
Hint: You do not need to give a solution in closed form.
Exercise 49: Use the power series method to find the solution of the initial value problem
y 00 (x) − 2xy 0 (x) − 2y(x) = 0,
y(0) = 1,
y 0 (0) = 0.
Determine the radius of convergence of the solution.
Exercise 50: Use the power series method to solve the following initial value problem
(2x − x2 )y 00 (x) + (1 − x)y 0 (x) = 0,
y(1) = 1,
y 0 (1) = 0.
Due date: Your written solutions are due on Tuesday, 1st July 2014. Please put them into the box in the
student office until 2:00 PM.
Tutorial No. 10
Advanced Mathematics II
Exercise T28:
ential equation
Calculate the characteristic polynomial of the following non-homogeneous third-order differy 000 (x) + y 00 (x) + 4y 0 (x) + 4y(x) = f (x) , x ∈ R .
For each of the following right hand sides f (x) = fi (x) determine a candidate for a particular solution using
the method of undetermined coefficients:
(a) f1 (x) = (4 + 12x)e2x ,
(b) f2 (x) = e−x ,
(c) f3 (x) = x2 cos(2x),
(d) f4 (x) = xe−x sin(2x).
Calculate particular solutions for f1 and f2 .
Exercise T29: Consider the following non-homogeneous second-order differential equation
y 00 (x) − 2y 0 (x) + 2y(x) = e2x sin x.
(a) Find a general real solution of the homogeneous differential equation.
(b) Apply the method of undetermined coefficients to find a particular solution.
(c) Solve the initial value problem given by y(0) = 53 , y 0 (0) = 1.
Exercise T30: Use the power series method to find a solution of the following differential equation
y 00 (x) − x y(x) = 0.
Find a recursive formula for the coefficients of the a solution depending on y(1) and y 0 (1) and calculate the first
four coefficients.
For detailed information regarding this course visit the following web page:
www.math.kit.edu/iag6/lehre/am22014s/en
Tutorial:
Friday, 27th June 2014, 9:45 AM