GATE Mechanical Notes-Applications of First Order Differential Equations

1) Growth and Decay problems

Let N(t) is the amount of substance (or population) that is either growing or decaying (decreasing). Let us say dN/dt is the time rate of change of amount of substance, and we assume that it is proportional to the amount of substance present, then dN/dt = kN, or

In the above equation, is the constant of proportionality. In these kind of questions we are assuming that N(t) is a differentiable and continuous function. Particularly for population problems, N(t) is a discrete and integer-valued, so the above assumption is not always applicable. 

2) Temperature problems

This set of problems uses Newton's law of cooling, which states that the time rate of change of the temperature of a body is proportional to the temperature difference between the body and its surrounding medium. Let T is the temperature of the body and let Tm is the temperature of the surrounding medium. Then the time rate of change of the temperature of the body is dT/dt, and Newton's law of cooling can be written as  or 

In the above equation, is a positive constant of proportionality. 

3) Falling body problems

Let us consider a vertically falling body of mass 'm' that is being influenced only by gravity 'g' and an air resistance that is proportional to the velocity of the body. We assume that the gravity and mass remain constant, and for the sake of convinience, downward direction is takn to be positive. 

As per Newton's second law of motion, the net force on the body is equal to the time rate of change of the momentum of the body, or for constant mass, , where F is the net force on the body and v is the velocity of the body, both at time t. 

For the falling body problem, there are two forces acting on the body: (1) the force due to gravity which is equal to the weight of the body, W = mg, and (2) the force due to air resistance which is -kv, where  is a constant of proportionality. The minus sign is needed as this force acts in the upward direction, therefore opposite to the direction of motion. So, the net force on the body is 

Upon substitution,  or , which is the equation of motion. 

If the air resistance is negligible, then k=0 and the above equation becomes, . When k>0 the limiting velocity  will be .

4) Dilution problems

To generalise it, let Q be the amount of salt in the tank at any time (t). The time rate of change of the amount of salt, dQ/dt is equal to the difference of rate at which salt enters tank and rate at which salt leaves the tank. 

To calculate the rate at which salt leaves the tank, calculate the volume of brine in the tank at any time (t), which is equal to the initial volume + volume of brine added (at) - volume of brine removed (bt), i.e. V + at - bt. 

The concentration of salt in the tank at any time is Q/(V+at-bt). From this it means that salt leaves the tank at the rate of b{Q/(V+at-bt)}.

 

or