Math assignment help
MATH 2255, Fall 2020 Homework 3
Due Friday, September 18, 11:30am.
Please upload your homework on Carmen. Late homework is not accepted. I encourage you to work with others on homework problems, but you must write up your own solutions. Solutions must be presented clearly, or will be marked down.
(1) Another equation that has been used to model population growth is the Gompertz equation
dy
dt = ry ln
( K
y
) ,
where r and K are positive constants. (a) Sketch the general solutions of this ODE by finding the critical points, and
determining the asymptotic stability of each. (b) For 0 ≤ y ≤ K, determine where the graph of y is concave up, and where it is
concave down. (c) For each 0 < y ≤ K, show that dy/dt as given by the Gompertz equation is
never less than dy/dt given by the logistic equation.
(2) Determine whether the following equations are exact. If they are, then solve the differential equations. (a) (2x + 3) + (2y − 2)y′ = 0; (b) dy
dt = −at+by
bt+cy ;
(c) (et sin y + 3y) − (3t− et sin y)y′ = 0; (d) (y/x + 6x) + (ln x− 2)y′ = 0, x > 0.
(3) Show that the given equations are not exact, but becomes exact when multiplied by the given integrating factor. Then solve the equation. (a) t2y3 + t(1 + y2)y′ = 0, M(t, y) = 1/ty3; (b) (t + 2) sin y + (t cos y)y′ = 0, M(t, y) = tet.
(4) Show that if (Nt−My)/M = Q(y) for some function Q, then the differential equation
M + Ny′ = 0
has an integrating factor of the form
M(y) = exp
(∫ Q(y)dy
) .
(5) Consider the initial value problem
y′ = 3 + t− y, y(0) = 1.
(a) Find the approximate values of the solution y(t) at t = 0.1, 0.2, 0.3 using the Euler method with h = 0.1;
(b) Repeat the above problem with h = 0.05, and compare your answers; (c) Find the solution y(t) and calculate y(t) at t = 0.1, 0, 2, 0.3, and compare your
values to your answers in parts (a) and (b).
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