Equilibrium solutions A differential equation of the form y′ ( t ) = f ( y ) is said to be autonomous ( the function f depends only on y ) . The constant function y = y 0 is an equilibrium solution of the equation provided f ( y 0 ) = 0 ( because then y′ ( t ) = 0 and the solution remains constant for all t ) . Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for t ≥ 0 . c. Sketch the solution curve that corresponds to the initial condition y (0) = 1 . 41. y ′ ( t ) = y ( y – 3)
Equilibrium solutions A differential equation of the form y′ ( t ) = f ( y ) is said to be autonomous ( the function f depends only on y ) . The constant function y = y 0 is an equilibrium solution of the equation provided f ( y 0 ) = 0 ( because then y′ ( t ) = 0 and the solution remains constant for all t ) . Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for t ≥ 0 . c. Sketch the solution curve that corresponds to the initial condition y (0) = 1 . 41. y ′ ( t ) = y ( y – 3)
Solution Summary: The author explains that the equilibrium solution of the given differential equation is y(t)=3 — the horizontal line segments represent equilibrium solutions in the directional fields.
Equilibrium solutionsA differential equation of the form y′(t) = f(y) is said to be autonomous (the function f depends only on y). The constant function y = y0 is an equilibrium solution of the equation provided f(y0) = 0 (because then y′(t) = 0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.
a.Find the equilibrium solutions.
b.Sketch the direction field, for t ≥ 0.
c.Sketch the solution curve that corresponds to the initial condition y (0) = 1.
41. y′ (t) = y (y – 3)
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
Are the functions f,g,f,g, and hh given below linearly independent?
f(x)=0, g(x)=cos(8x), h(x)=sin(8x)
If they are independent, enter all zeroes. If they are not linearly independent, find a nontrivial solution to the equation below. Be sure you can justify your answer.
Are the functions f, g, and h given below linearly independent?
f(x) = e2" + cos(7x), g(æ)= e2 – cos(7x), h(x) = cos(7x).
If they are independent, enter all zeroes. If they are not linearly independent, find a nontrivial solution to the equation below. Be sure you can justify your answer.
(e2z + cos(7x)) +
(e2z – cos(7x)) +
(cos(7x)) = 0. help (numbers)
y" +p (t)y' +q(t)y = 0 (i),
can be put in more suitable form for finding a solution by making a change of the independent and/or dependent variables. Determine conditions on p and q that enable this equation to be transformed into an equation with constant coefficients by a change of the independent variable. Let r = u (t) be the new independent variable. It is easy to show
(島)
( + ( +p(t) + a()y = 0 (1i).
dy
that
dt
dz dy
d'y
and
2 d'y
d²z dy
dt dz
The differential equation becomes
dt dr
dt
d'y
d'z
de
dz dy
d'y
, 9 and y must be proportional. If g (t) > 0, then choose the constant of proportionality to be 1. Hence, x = u (t) = [g (t)]i dt. With a chosen this way, the coefficient of in equation (ii) is also a constant, provided that the function H =
g'(t) +2p(t)q(t)
is a
dy
In order for equation (ii) to have constant coefficients, the coefficients of
constant. Thus, equation (i) can be transformed into an equation with constant coefficients by a change of the independent variable,…
Thomas' Calculus: Early Transcendentals (14th Edition)
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