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Homework 6: Math 22B

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Homework 6: Math 22B
(20 points)
Instructions : Solve all problems. Print out your solutions when computer results are asked for,
work neatly, label your plots, show your work. Staple your homework together with your name
on it. A random subset of the problems will be graded. You are encouraged to work in groups,
but everyone must do their own write up.
Warning : Unstapled homework with multiple pages is minimum -5 out of 20 points and if a page
is lost from an unstapled homework the default assumption will be it was not turned in.
Reading Assignment : Read Boyce and Diprima Chapter 6.1 – 6.2
Problem 1 : Find the general solution using the method of undetermined coecients (3.5.5)
y00 + 9y = t
2
e3t + 6
Problem 2 : Find the general solution using the method of undetermined coecients (3.5.8)
y00 + 2y0 + y = 2et
Problem 3 : Find the general solution using the method of undetermined coecients (3.5.18)
y00 + 2y0 + 5y = 4et cos(2t), y(0 = 1, y0
(0) = 0
Problem 4 : (3.6.1) Use the variation of parameters method from section 3.6 to solve for the general
solution
y00 5y0 + 6y = 2et
Check your answer by the method of undetermined coecients.
Problem 5 : (3.6.15) Verify that y1 =1+ t and y2 = et satisfy the di↵erential equation
ty00 (1 + t)y0 + y = t
2
e2t
, t 0
then find the general solution using the variation of parameters method to find the particular
solution.
Problem 6 : (3.7.1) Find R, !0, to write u = 3 cos(2t) + 4 sin(2t) = R cos(!0t ).
Problem 7 : (3.7.2) Find R, !0, to write u = 2 cos(⇡t) 3 sin(⇡t) = R cos(!0t ).
Problem 8 : (3.7.26) The position of a certain spring satisfies
mu00 + u0 + ku = 0, u(0) = u0, u0
(0) = v0
Solve the initial value problem, assuming 2 < 4km. Then determine R in terms of m, , k, u0, v0
and write the solution in the form
u(t) = Ret/2m cos(µt )
Problem 9 : (3.7.29) The position of a certain spring satisfies
u00 +
1
4
u0 + 2u = 0, u(0) = 0, u0
(0) = 2
(a) Find the solution and (b) using MATLAB plot u vs. t and u0 vs. t on the same set of axis.
Problem 10 : (3.8.19) Consider the vibrating system
u00 + u = 3 cos(!t), u(0) = 1, u0
(0) = 1
(a) Find the solution for ! 6= 1, (b) using MATLAB plot u vs. t for ! = 0.7.