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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.


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.


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}.


Prove a substitution converts a given second order equation to a first order equation

  1. Consider the second-order differential equation

        \[ y'' + P(x) y' + Q(x) y = 0 \]

    and let u be a solution to the equation. Show that the substitution y = uv converts the equation

        \[ y''  + P(x)y' + Q(x)y = R(x) \]

    into a first-order liner equation for v'.

  2. By inspection, find a nonzero solution of the second order differential equation

        \[ y'' - 4y' + x^2 (y' - 4y) = 0 \]

    and use part (a) to find a solution of the differential equation

        \[ y'' - 4y' + x^2(y' - 4y) = 2xe^{-\frac{x^3}{3}} \]


        \[ y = 0 \quad \text{and} \quad y' = 4 \quad \text{when } x = 0. \]