Find the general solution to y′′+6y′+13y=0. Give your answer as y= In your answer, use c1 and c2 to denote arbitrary constants and x the independent variable. Enter c1 as c1 and c2 as c2.

Step-by-step explanation:

In order to find the general solution of a homogeneous second order differential equation, we need to solve the characteristic equation. This is basically as easy as solving a quadratic.

For a second order differential equation of type:

Has characteristic equation:

Whose solutions are the roots from which the general solution can be formed. There are three cases:

Real roots:

Repeated roots:

Complex roots:

Therefore:

The characteristic equation for:

Is:

Solving for :

So:

Hence, the general solution of the differential equation will be given by:

Survey, observational study, observational study, survey

Step-by-step explanation:

In a observational study, the sample is only observed.  No variables are controlled.In a survey, information is obtained directly from the sample, using a questionnaire to measure opinions.In an experiment, variables are controlled by design to study a link of causation.

In the first one, the director wants to know what his singers do before a performance.  The director cannot observe each singer at the same time, so he should use a survey.

In the second one, the director wants to know who his best soprano is.  The best way to determine this is by simply observing.  Observational study.

In the third one, the directer wants to know how verbally disciplining a singer affects the group.  Again, the best way to determine this is by observing.  Observational study.

In the fourth one, the director wants to know what musical is the most popular.  This of course requires a survey.