Differential Equations and Mathematical Models

Differential Equations and Mathematical Models

1. Separable

   • Newton's Law of Cooling and Heating
     • dT/dt = k (A - T)
   • Torricelli's Law
     • dV/dt = -k [(y)^(1/2)]
   • Natural Growth
     • Equation: P(t) = Po*[ e^(kt)]
     • (decay) dx/dt = kt
     • Change of Population: dP/dt = kP
       • Logistic Equation
         • dP/dt = kP(M-P)
         • Joint Proportion
         • Limited Environment
         • Competition
         • P(t) = MPo / [Po + (M-Po)^(-kMt)]
   • Radioactive Decay
     • dN/dt = -kN
   • Drug Elinination
     • dA/dt = -λA
   • Velocity and Acceleration
     • v = dx/dt
     • a = dv/dt = d^2x/dt^2

2. Linear

   • A tank containing a solution (amount of solute)
     • dx/dt = ri*ci - (ro/v)x
   • Indoor Temperature Oscillation
     • A(t) = a0 + a1cos(wt) + b1sin(wt)

3. Homogeneous

   • Flight Trajectories
     • y(x) = (a/2)*{[(x/a)^(1-k)] - [(x/a)^(1+k)]}

4. Autonomous

   • Harvesting a logistic population
     • dx/dt = ax - bx^2 - h