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1. (a) On what interval are we guaranteed (via Theorem 4.1.1) that the initial value problem
dy + 2y =0
'(0) = 0
has a unique solution?
(b) Without referring to the general solution below, what is the particular solution to the IVP in (a)
Provide your reasoning as well as the particular solution.
(c) The general solution to the differential equation - 242 + 2y -0 is
y = cje" cost + cge" sins.
Use the general solution to describe all solutions to the following Boundary Value Problem
day _ 24+2y =0
( ( ) -0
The conditions y(0) = 0:and y(#) = 0 are called boundary conditions as we are looking for
solutions that satisfy conditions on the boundary of the interval (0, =].
2. Consider the differential equation
(a) Verify that Vi - c' is a solution to ().
(b) Use reduction of order to produce a second, linearly Independent solution ya to (=).
(c) Verify that in and yr are linearly independent.
(d) Do the functions y; and y, form a fundamental set for (.)? Why or why not?
(e) Describe the general solution to (.).
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