Calculus
# First Order Differential Equations

Two forces act on a parachutist. One is \(mg,\) the attraction by the earth, where \(m\) is the mass of the person plus equipment and \(g=9.8 \text{ m/sec}^2\) is the acceleration of gravity. The other force is the air resistance ("drag"), which is assumed to be proportional to the square of the velocity \(v(t)\).

Using Newton's second law of motion (mass \(\times\) acceleration = net force applied), set up an ordinary differential equation for \(v(t).\)

Let \(k\) denote the drag coefficient.

You are given the differential equation \( \frac{dy}{dx} + y \ln x = x^{-x} e^{-x} \) and the initial condition \( y(1) = 0 \)

If the value of \( y(2) \) can be written in the form \( \dfrac{e^{2}+a}{be^2} \) determine the value of \( a + b \).

**order** of the differential equation \[x^2.\left( \dfrac{d^2y}{dx^2} \right)^6+y^{\frac{-2}{3}}.\sqrt{1+\left( \dfrac{d^3y}{dx^3} \right)^5}+\dfrac{d^2}{dx^2}. \left( \dfrac{d^2y}{dx^2} \right)^{\frac{-2}{3}} =0?\]

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