Differential Equations Answers

1. Solve the differential equation. Use the fact that the given equation is homogeneous.
dy / dx = (x^2 + 8y^2) / 3xy

2. A certain college graduate borrows 7486 dollars to buy a car. The lender charges interest at an annual rate of 19% . Assuming that interest is compounded continuously and that the borrower makes payments continuously at a constant annual rate k dollars per year, determine the payment rate that is required to pay off the loan in 2 years. Also determine how much interest is paid during the 2-year period.

Round your answers to two decimal places.

A. Payment rate ______ dollars per year. (absolute tolerance +/- 0.01)
B. Interest Paid _____ dollars (absolute tolerance is +/- 0.05)

1. Find the solution of the given initial value problem.
y prime - y = 3t(e^2t), y(0) = 1

y(t) = _____

2. Find the solution of the given initial value problem in explicit form.

y prime = (1-11x)y^2, y(0) = -1/6

Enclose numerators and denominators in parentheses. For example, (a-b) / (1+n).

y(x) = _____

1. A skydiver weighing 174 lbf (including equipment) falls vertically downward from an altitude of 4000 ft and opens the parachute after 10s of free fall. Assume that the force of air resistance, which is directed opposite to the velocity, is 0.76 |v| when the parachute is closed and 12|v| when the parachute is open, where the velocity v is measured in ft/s.

Measure the positive direction of motion downward. Consider that x(0) = 0 ft. Use g = 32 ft/s^2. Round your answers to two decimal places.

(a) Find the speed of the skydiver when the parachute opens. (the tolerance is +/- 2 percent)
v(10) = _______ ft/s

(b) Find the distance fallen before the parachute opens.(the tolerance is +/- 2 percent)
x(10) = _________ft

(c) What is the limiting velocity vl after the parachute opens?(the tolerance is +/- 2 percent)
vl = _____ ft/s

1. A given field mouse population satisfies the differential equation dp/dt = 0.5p-330 where p is the number of mice and t is the time in months.

(a) Find the time at which the population becomes extinct if p(0)=620. Round your answer to two decimal places. The absolute tolerance +/- 0.01.

tf = ____________ (months)

(b) Find the initial population p0 if the is to become extinct in 1 year. Round your answer to the nearest integer. The absolute tolerance +/- 0.01.

p0= ________ (mice)

The rate of increase of susceptible AIDS victims is proportional to the number of susceptible
persons and number of infected persons. If there are 0
S susceptible persons and 1 infected
person at a time to then i) set up the equation for the spread of the disease ii) solve the resulting
equation iii) give a physical interpretation to the same by plotting the epidemic curve iv) write the
limitations of the model.

sove pde d²z/dx²+d²z/dxdy+dz/dy-Z=e^-x

z = f(x + ay) + g(x − ay)

The rate of change of the value of an investment, S, with respect to time, t ≥ 0, is given by dS dt =1000r 10ert/100, where r is the annual interest rate (assumed constant) and the principal of the investment is S(0) = 10 000. (a) Find an expression for S(t), that is, the value of the investment at time t. (6 marks) (b) Verify that your expression for S(t) is correct by computing S(t). (1 mark) (c) Explain why S(t) is continuous for t ≥ 0. (1 mark) (d) Determine, by computing lim t→∞S(t), what would happen to the value of the investment if t were to grow without bound. Interpret the result. (5 marks) (e) How long would it take for the value of the investment to be exactly 15 000? (4 marks)

Solve the following differential equation by changing the independent variable:
x.d^2y/dx^2+(4x^2-1)dy/dx+4x^3=2x^3

solve the following initial value problem y" + y' -2y = -6sin2x-18cos2x y(0)=2,y'(0)=2