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First Order Differential Equations

These are equations, Calculus-style. From modeling real-world phenomenon, from the path of a rocket to the cooling of a physical object, Differential Equations are all around us.

Formulate a Statement

Suppose that we drop a stone, which falls freely with no air resistance. Experiments show that, under that assumption of negligible air resistance, the acceleration \( {y}^{\prime\prime}=\frac { {d}^{2}y }{ {d}^{2}t } \) of this motion is constant, i.e. equal to the so-called acceleration of gravity \(g=9.8 \text{ m/sec}^2.\) State this as an ordinary differential equation for \(y(t),\) the distance fallen as a function of time \(t.\)

Because of limited food and space, a squirrel population cannot exceed \(2900\) squirrels. The population grows at a rate proportional to the product of the existing population and the attainable additional population. If \(P\) denotes the squirrel population at time \(t,\) which of the following equations represents the population growth rate for \(k>0?\)

In psychology, a stimulus-response situation is a situation in which the response \(y=f(x)\) changes at a rate inversely proportional to the strength of the stimulus \(x\). Which of the following equations represents this?

For a body moving along a straight line, let \(y(t)\) denote its distance from a fixed point \(O\) at time \(t.\) Given that velocity plus distance is equal to square of time, find the ordinary differential equation of the motion.

Jack has a magic beanstalk, whose rate of growth is directly proportional to its height \(H\). Which of the following equations represents his beanstalk's height as a function of time \(t\)?

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