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Discuss properties of the solution of a differential equation

Consider the differential equation

    \[ y' = \frac{2y^2 + x}{3y^2 + 5}. \]

Let y = f(x) be a solution to the equation with the initial condition f(0)= 0. Without attempting to solve this equation explicitly answer the following questions.

  1. Since f(0)= 0 we also have f'(0) = 0. Does f have a relative maximum, relative minimum, or neither at 0?
  2. If x \geq 0 then f'(x) \geq 0 and if x \geq \frac{10}{3} then f'(x) \geq \frac{2}{3}. Find positive numbers a and b such that

        \[ f(x) > ax - b \qquad \text{for each } x \geq \frac{10}{3}. \]

  3. Prove that

        \[ \lim_{x \to +\infty} \frac{x}{y^2} = 0. \]

  4. Prove that

        \[ \lim_{x \to +\infty} \frac{y}{x} = A \]

    for some finite number A and find the value of A.


Incomplete.

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

The population of a town at time t = 0 is 365. The population growth factor is e, 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 t 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 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 t years.
  2. Find the decay constant of the material in units \operatorname{gm}^{-1} \operatorname{yr}^{-1}.

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 \left( \frac{25}{24} \right)^2 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 f(x) be a nonnegative, differentiable function whose graph passes through both points (0,0) and \left( 1, \frac{2}{\pi}\right). For every real number x > 0, the ordinate set of f on the interval [0,x] generates a solid of revolution when rotated about the x-axis whose volume is given by

    \[ x^2 f(x). \]

Find the formula for the function f(x).


Incomplete.

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

Consider a curve whose Cartesian equation is given by y = f(x), 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 f(x). The curve f(x) then divides the rectangle into two pieces A and B. These two pieces of the rectangle then generate solids of revolution when rotated about the x-axis. If the volume of one solid of revolution is always n times the volume of the other solid of revolution, find the equation for f(x).


Incomplete.

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

Consider a curve whose Cartesian equation is given by y = f(x), 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 f(x). The curve f(x) then divides the rectangle into two pieces A and B such that one of the regions has area n times the area of the other for every such rectangle. Find the equation of f(x).


Incomplete.