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Homework 5: Math 22B

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Homework 5: 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 3.5-3.9
Problem 1 : [Programming in Matlab – section 2.7] Write a simple program in Matlab to solve the
initial value problem for t ∈ [0, 5].
dy
dt = −ty + 0.5y
3
, y(0) = 1
using the Euler Method and an appropriately chosen time step. Make sure to include the code
with your homework.
Problem 2 : Solve the given IVP and use Matlab to plot your solution (just plotting, you don’t
have to solve it numerically).
y
00 + 3y = 0, y(0) = −2, y0
(0) = 3
Problem 3 : Solve the IVP
2y
00 + 3y
0 − 2y = 0, y(0) = 1, y0
(0) = −β
and (b) plot the solution when β = 1 and find the coordinates (t0, y0) of the minimum point of
the solution in this case.
Problem 4 : Verify that y1 = e
t and y2 = tet are solutions of y
00 − 2y
0 + y = 0. Do they form a
fundamental set of solutions?
Problem 5 :
Problem 6 : Use the method of order reduction to find a second solution given y1 = t
2
, t 0 and
t
2
y
00 − 4ty0 + 6y = 0
Problem 7 : Use the method of order reduction to find a second solution given y1 = e
x
, x 1 and
(x − 1)y
00 − xy0 + y = 0
Problem 8 :Find the general solution to y
00 − 2y
0 − 10y = 0
Problem 9 : Find the particular solution
y
00 + 2y
0 + 2y = 0, y(π/4) = 0, y0
(π/4) = −2
Problem 10 : Find the particular solution
y
00 − 6y
0 + 9y = 0, y(0) = 0, y0
(0) = 2