Differential Equations Answers

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)

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

The function f(x,y) = max {y/x,x} is a homogeneous function on R^2

P(x)y"+a(x)y'=0