Differential Equations Answers

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

Solve the following B.V.P.

𝑢𝑡𝑡 = 𝑐 2𝑢𝑥𝑥; { 𝑢(𝑥, 0) = cos 𝑥 − 1, 𝑢𝑡 (𝑥, 0) = 0 𝑢(0,𝑡) = 0, 𝑢(2𝜋,𝑡) = 0 0 ≤ 𝑥 ≤ 2�

A string of length 10 𝑓𝑡. is raised at the middle to a distance of 1 𝑓𝑡, and then released. Describe the motion of the string assuming 𝑐 = 1𝑓𝑡. 𝑠 −1. 

The ends and sides of a thin copper bar (𝛼 2 = 1.14) of length 2 are insulated so that no heat can pass through them. Find the temperature 𝑢(𝑥,𝑡) in the bar if initially

𝑢(𝑥, 0) = { 60𝑥 0 < 𝑥 < 1 60(2 − 𝑥) 1 ≤ 𝑥 < 2 

solve the system of first-order linear differential equations

y1'=y1-y2

y'2=2y1+4y2