Applications of Differential Equations

Rule to find the equation of orthogonal trajectories

Note: Differentiate as many times as there are constants.

For rectangular cartesian co-ordinates:

Step 1: Given f(x,y,a) = 0, where a is a variable parameter.

Step 2: Differentiate f(x,y,a) = 0 w.r.t. x and eliminate ‘a’.

Step 3: Replace ${dy}/{dx}$ by $- {dx}/{dy}$

For polar co-ordinates:

Step 1: Given f(r,θ,a) = 0, where a is a variable parameter.

Step 2: Differentiate f(r,θ,${dr}/{dθ}$) = 0 w.r.t. x and eliminate ‘a’.

Step 3: Replace ${dr}/{dθ}$ by ($- r^2{dθ}/{dr}$)

Rate of decay of radioactive materials

If u is amount of material at any time t,

${du}/{dt}$ = – ku , where k is constant

Newton’s law of cooling

If $θ_o$ is the temperature of the surroundings and θ that of the body at any time t, then

${dθ}/{dt}$ = – k (θ – $θ_o$) , where k is constant

Rectilinear motion

Velocity (v): ${dx}/{dt}$

Acceleration (a): ${dv}/{dt}$ or ${d^2 x}/{dt^2}$ or $v {dv}/{dx}$

Newton’s second law of motion: F = $d/{dt} (mv)$ , where F is effective force

D’Alembert’s principle: Net force = Mass x Acceleration

Note: When resistance becomes equal to the weight, the acceleration becomes zero and particle continues to fall with a constant velocity, called the limiting or terminal velocity.