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.

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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