Differential Equations Answers

1. If a wet sheet in a dryer loses its moisture at a rate proportional to its moisture content, and if it loses half of its moisture during the first 10 min of drying, when will it be practically dry, say, when will it have lost 99% of its moisture?

Find the general solution to the given partial differential equation and use it to find the solution satisfying the given initial data.

"\\frac{\\partial u}{\\partial x}=-(2x+y)\\frac{\\partial u}{\\partial y}"

"u(0,y)=1+y^2"

(10-6y+e^(-3x))dx-2dy=0 

x + 2(xp − y) + p^2 = 0

dx/(z^2+2y)=dy/(z^2+2x)=dz/-z

find d/dt (F-G) if if F(t) = 4t^2i - tj + t^2k and G(t) = ti + 2t^2j + 4 sin t k

2. *A body is released from rest and moves under uniform gravity in a medium that exerts a resistance force proportional to the square of its speed and in which the body’s terminal speed is V . Show that the time taken for the body to fall a distance h is

V/g cosh−1 e^(gh/v^2) .

In his famous (but probably apocryphal) experiment, Galileo dropped different objects from the top of the tower of Pisa and timed how long they took to reach the ground. If Galileo had dropped two iron balls, of 5 mm and 5 cm radius respectively, from a height of 25 m, what would the descent times have been? Is it likely that this difference could have been detected? [Use the quadratic law of resistance with C = 0.8. The density of iron is 7500 kgm−3.]

Use linear substitution to solve the following first-order differential equation

𝑑𝑦/𝑑𝑥=(2𝑥+𝑦)/(2𝑥+𝑦+1)

dx/(z^2+2y)=dy/(z^2+2x)=dz/-z

4/7x+5/14x=39