A differential equation of the form y prime (t )equalsF(y) is said to be autonomous (the function F depends only on y). The constant function yequals y 0 is an equilibrium solution of the equation provided Upper F (y 0 )equals0 (because then y prime (t )equals0, and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal line segments in the direction field. Note also that for autonomous equations, the direction field is independent of t. Consider the equation y prime (t )equals3 sine y .

a. Find all equilibrium solutions in the interva

Question

contestada

Respuesta :

josealejandrocl josealejandrocl · Answer 1 · 2020-01-04T17:34:44+01:00

Answer:

y=n*pi, where n is an integer.

Step-by-step explanation:

We have the equation:

[tex]y'(t)=3sin(y)[/tex]

Which is autonomous because the independent variable does not appear explicitly.

Now, we are asked about all the equilibrium solutions for such autonomous equation. Let's remember that the solution equilibrium y=y0 when F(y0)=0.

Then, matching the given equation to zero, we have:

[tex]y'(t)=3sin(y)=0[/tex]

Which is fulfilled when the sine function has a value of zero. The sine function is worth zero in y=n*pi where n is an integer.