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Discuss properties of the solution of a differential equation

Consider the differential equation

Let be a solution to the equation with the initial condition . Without attempting to solve this equation explicitly answer the following questions.

1. Since we also have . Does have a relative maximum, relative minimum, or neither at 0?
2. If then and if then . Find positive numbers and such that

3. Prove that

4. Prove that

for some finite number and find the value of .

Incomplete.

Find the launch velocity necessary for a rocket to escape Earth only considering gravity

Ignoring all forces except the Earth’s gravity find the speed with which you must launch a rocket so that it never returns to Earth.

Incomplete.

Find and solve a differential equation governing population growth with given conditions

The population of a town at time is 365. The population growth factor is , and the town experiences a death rate of one percent of the population per day. Find a differential equation modeling the population of the town as a function of time and find

1. the actual population of the town after years,
2. the cumulative total of the fatalities from the town’s death rate.

Incomplete.

Find and solve a differential equation for the decay of a material with given properties

A given substance decays at a rate proportional to the squareroot of the amount of the material present. Show that a substance with this decay property will completely decay in a finite amount of time and find this time.

Incomplete.

Find and solve a differential equation for the decay of a material with given properties

A given substance decays at a rate proportional to the square of the amount of the material present. At the end of one year there is 0.5 grams of the substance remaining.

1. Create and solve a differential equation that governs the mass of the material present after years.
2. Find the decay constant of the material in units .

Incomplete.

Find the time until a tank with a hole is empty

Consider a water tank shaped like a right circular cone with its point up. Find the amount of time necessary for the tank to completely drain if there is a hole in its base. Express the time as a function of the dimension of the water tank and the area of the hole in the tank.

Incomplete.

Find the depth of water in a leaky tank as time goes to infinity

Consider a water tank with vertical sides and with cross-section a square of area 4 feet. Water exits the tank through a hole which as area equal to \frac{5}{3} square inches, and water is added to the tank at a rate of 100 cubic inches per second. Show that the water level approaches the value of feet above the hole no matter what the initial water level was.

Incomplete.

Find a function whose ordinate set generates a solid of revolution with volume x2 f(x)

Let be a nonnegative, differentiable function whose graph passes through both points and . For every real number , the ordinate set of on the interval generates a solid of revolution when rotated about the -axis whose volume is given by

Find the formula for the function .

Incomplete.

Find a function which divides a rectangle into pieces with given properties

Consider a curve whose Cartesian equation is given by , and which passes through the origin. A rectangular region is drawn with one corner at the origin, and the other corner on the curve of the graph of . The curve then divides the rectangle into two pieces and . These two pieces of the rectangle then generate solids of revolution when rotated about the -axis. If the volume of one solid of revolution is always times the volume of the other solid of revolution, find the equation for .

Incomplete.

Find a function which divides a rectangle into pieces with given properties

Consider a curve whose Cartesian equation is given by , and which passes through the origin. A rectangular region is drawn with one corner at the origin, and the other corner on the curve of the graph of . The curve then divides the rectangle into two pieces and such that one of the regions has area times the area of the other for every such rectangle. Find the equation of .

Incomplete.