Each of these ODEs is linear and homogeneous with constant coefficients, so we only need to find the roots to their respective characteristic equations.
(a) The characteristic equation for
is
which arises from the ansatz 
.
The characteristic roots are 
and 
. Then the general solution is
where 
are arbitrary constants.
(b) The characteristic equation here is
with a root at 
of multiplicity 2. Then the general solution is
(c) The characteristic equation is
with roots at 
, where 
. Then the general solution is
Recall Euler's identity,
Then we can rewrite the solution as
or even more simply as
Hey there
<span>The first step is to carefully read every single word of the question and answers,keeping in mind that the some small detail can affect the answer.
You should also pay great attention to negative words and prefixes. Words such as "un" can make your answer wrong if unread properly. If an event is "unlikely to happen," that means it is most likely an occurrence.</span>
P represents heating and Q represent cooling.
this is because when you start from a liquid and turn to a gas the particles move faster making them farther apart, causing a gas. the opposite happens when you go from a liquid to a solid. the partials move closer together causing a solid to form. :)
