Hi I am having difficulty with these differential equation problems.
1. A tank contains ? kg of salt and ? L of water. Pure water enters a tank at the rate ?
L/min. The solution is mixed and drains from the tank at the rate ?
a. Find the amount of salt in the tank after 3 hours.
2. The general solution to the second?order differential equation d2ydt2?2dydt+10y=0 is in
the form y(x)=e?x(c1cos?x+c2sin?x). Find the values of ? and ?, where ? > 0.
3. A brick of mass 8 kg hangs from the end of a spring. When the brick is at rest, the spring
is stretched by 3 cm. The spring is then stretched an additional 4 cm and released.
Assume there is no air resistance. Note that the acceleration due to gravity, g, is g=980
cm/s2. Set up a differential equation with initial conditions describing the motion and
solve it for the displacement s(t) of the mass from its equilibrium position (with the
spring stretched 3 cm).
4. Suppose a spring with spring constant 7 N/m is horizontal and has one end attached to a
wall and the other end attached to a 2 kg mass. Suppose that the friction of the mass with
the floor (i.e., the damping constant) is 3 N?
a. Set up a differential equation that describes this system. Let x to denote the
displacement, in meters, of the mass from its equilibrium position, and give your
answer in terms of x,x?,x??. Assume that positive displacement means the mass is
farther from the wall than when the system is at equilibrium.
b. Find the general solution to your differential equation from the previous part. Use
c1 and c2 to denote arbitrary constants. Use t for independent variable to represent
the time elapsed in seconds. Enter c1 as c1 and c2 as c2.
c. Enter a value for the damping constant that would make the system critically