Uranium disintegrates at a rate proportional to the amount present at any time. If m₁ and m₂ grams of uranium are present at time t₁ and t₂ respectively then show that the half life of uranium is  

In a culture of bacteria the rate of increase is proportional to the number present. If their number is 3000 and 5000 at the end of 3 and 4 hours, find the number in the beginning.

A tank is initially filled with 100 litres of salt solution containing 0.1 kg of salt per litre. Fresh brine containing 0.2 kg of salt per litre runs into the tank at the rate of 5 litres / minute and the mixture, assumed to be kept uniform by stirring, runs out at the same rate. Find the amount of salt at any time and determine how long will it take for this amount to reach 15 kg.

Newton’s law of cooling states that the rate of cooling of a body is proportional to the difference of temperature of the body and its surrounding. Assuming u₁ is the original temperature of the body (at  t = 0) and that of the surrounding u₀, find the temperature u of the body at time t.

The rate of cooling of a heated body is proportional to the difference in the temperature between the body and the surrounding medium. If the temperature of the body drops from 100°C to 60°C in one minute when the temperature of the surrounding is 20°C, what will be the temperature of the body at the end of the second minute?

Find the equation of the curve through the origin which satisfies the differential equation  

Find the differential equation of the system of circles touching the Y–axis at the origin.

Find the differential equation satisfied by the system of parabolas y² = 4a (x + a) and show that its orthogonal trajectory of the system belongs to the system itself.

Solve  

Solve  

Solve  

Solve cos (x + y) dy = dx

Solve  

Solve  

Solve  

Solve  

Solve  

Solve  

Show that the curve in which the angle between the tangent and the radius vector at any point is one-half of the vectorial angle, is a cardioid.

The slope of a curve at every point is the reciprocal of twice the ordinate at that point. The curve also passes through the point

(4, 3). Find the equation of the curve.

Solve  (y²e^x + 2xy) dx – x²dy = 0

A hundred gallon tank is filled with brine containing 60 pounds of dissolved salt. Water runs into the tank at the rate of 2 gallons per minute and the mixture, kept uniform by stirring, runs out at the same rate. How much salt is in the tank after one hour?

The air in certain room 150’ x 50’ x 12’ tested 0.25 CO₂. Fresh air containing 0.05% CO₂ was then admitted by ventilator at the rate of 9000 ft³/min. Find the percentage of CO₂ after 20 minutes?