Applications of Differential Equations
December 8, 2016
Categorised in: 1st Year Maths 2
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.
Simple electric circuits
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