Calculus

First Order Differential Equations

First Order Differential Equations: Level 2 Challenges

         

Two forces act on a parachutist. One is mg,mg, the attraction by the earth, where mm is the mass of the person plus equipment and g=9.8 m/sec2g=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)v(t).

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

Let kk denote the drag coefficient.

You have a faulty laptop battery whose charge gauge acts very strangely. Particularly, after every minute, its estimate of how much time you have left before it dies drops by dd minutes, where dd is a positive number. The gauge estimates the rate at which you deplete the battery by finding the average rate of change of your battery charge thus far with respect to time. Your battery takes exactly mm minutes to die. The function f(t)f(t), where tt is a positive number of minutes, describes the amount of battery charge used (i.e. f(0)=0f(0) = 0 when your battery is charged, and f(m)=1f(m) = 1 when your battery is dead). Which is the correct form of f(t)f(t)?

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

If the value of y(2) y(2) can be written in the form e2+abe2 \dfrac{e^{2}+a}{be^2} determine the value of a+b a + b .

Solve the ODE xy+y=(xy)2lnxxy'+y={(xy)}^{2} \ln{x} where y(1)=13y(1)=\frac{1}{3}. Calculate the value of y(e)y(e) to 3 decimal places.

What is the order of the differential equation x2.(d2ydx2)6+y23.1+(d3ydx3)5+d2dx2.(d2ydx2)23=0?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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