Differential Equations

Solve the differential equation: \(x(y-1)dx+(x+1)dy=0\). If \(y=2\) when \(x=1\), determine \(y\) when \(x=2\).

Solve \((\cos x \cos y-\cot x)dx-\sin x \sin y\,dy=0\).

Solve \((y-\sqrt{x^2+y^2})dx-x\,dy=0\).

Solve \((x+y)dy=(x-y)dx\).

An object falls from rest in a medium offering a resistance. The velocity of the object before the object reaches the ground is given by the differential equation \(dV/dt+V/10=32\), ft/sec. What is the velocity of the object one second after it falls?

In a tank are 100 liters of brine containing 50 kg. total of dissolved salt. Pure water is allowed to run into the tank at the rate of 3 liters a minute. Brine runs out of the tank at the rate of 2 liters a minute. The instantaneous concentration in the tank is kept uniform by stirring. How much salt is in the tank at the end of one hour?

A tank initially holds 100 gallons of salt solution in which 50 lbs of salt has been dissolved. A pipe fills the tank with brine at the rate of 3 gpm, containing 2 lbs of dissolved salt per gallon. Assuming that the mixture is kept uniform by stirring, a drain pipe draws out of the tank the mixture at 2 gpm. Find the amount of salt in the tank at the end of 30 minutes.