Differential Equation

Radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100 mg of radium decomposes to 96 mg. How many mg will be left after 100 years?

The population of a country doubles in 50 years. How many years will it be five times as much? Assume that the rate of increase is proportional to the number of inhabitants.

Radium decomposes at a rate proportional to the amount present. If half of the original amount disappears after 1000 years, what is the percentage lost in 100 years?

Find the equation of the family of orthogonal trajectories of the system of parabolas \(y^2=2x+c\).

According to Newton's law of cooling, the rate at which a substance cools in air is directly proportional to the difference between the temperature of the substance and that of air. If the temperature of the air is \(30^\circ\) and the substance cools from \(100^\circ\) to \(70^\circ\) in 15 minutes, how long will it take to cool \(100^\circ\) to \(50^\circ\)?

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.

If the nominal interest rate is 3%, how much is P5,000 worth in 10 years in a continuously compounded account?

A nominal interest of 3% compounded continuously is given on the account. What is the accumulated amount of P10,000 after 10 years?