Differential Equations Answers

1- Suppose a body weighing 64 Ibs attached to a spring stretches It by 8 feet . Assume that damping force, numerically equal to 8 times the instantaneous velocity acts on the mass .In addition an external force f(t)=6t2 4 is being applied to the system .At t=0 the body is released from rest at appoint 3 feet above the equilibrium position.
a- Setup a mathematical model to find displacement of the body from equilibrium position at time t.
b- Solve the model analytically and find a formula for the displacement at any time t.
C- Draw simulation block diagram.

The rate at which a population of lions at etosha national park P(t) is progressing is given by the differential equation
dp/dt=P(M-kP) where M, k are positive constants
Solve the differential equation to determine an expression for P(t)

solve the initial value problem x_dx^dy-2y=〖2x〗^4, y(2)=8

suppose the population P(t) of Ongwediva town at any time t>0 satisfies the logic law
dP/dt=P/100-P^2/〖10〗^8
Where t is measured in years. Given that the population of this city was 100000 in 1980 determine the population as a function t>1980

if the population S(t) of a species of mosquitos in the Okavango region Delta at any time t >0 is modeled by the differential equation θ(ds/dt)+γS=〖αe〗^(-φt), whereθ,γ and α are positive constants and φ is a nonnegative constant, and if the initial population of this species is S(0)=S_0,
show that the mosquito population S(t) at any time t >0 is given by S(t)=α/(γ-θφ) e^(-φt)+(S_0-α/(γ-θφ) ) e^(-φt)

show that if φ=0 the species population S(t) approaches α/γ as t →∞ but if φ> the S(t) approaches zero as t→∞

find linearly independent function that are annihilated by the given differential operator D⁵,(D²+4D)

Find the integral of (y² -1)dx -2dy =0

The taut string of length 50 cm fastened at both ends, is distributed from its position of equilibrium by imparting to each of its points an initial velocity of magnitude kx for 0<x<50. Find the displacement function y(x, t).

The taut string of length 50 cm fastened at both ends, is distributed from its position of equilibrium by imparting to each of its points an initial velocity of magnitude kx for 0<x<50. Find the displacement function y(x, t).