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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.

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