Applications of differential equation in Physics and Biology

Applications of differential equation in Physics and Biology

• 1.
  APPLICATIONS OF DIFFERENTIAL EQUATIONIN PHYSICS Group - IV
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  ◦ Differential equationsare broadly used in all the major scientific disciplines such as physics, chemistry and engineering. ◦ Solving the reduced, specific differential equation provides the solution that fully describes the processes and phenomena being studied, where they are so complex that they otherwise are not obvious without the solution to the differential equation. ◦ The insight gained by solving the differential equation(s) leads to a more complete understanding of the physical processes being studied. ◦ Differential equations are the foundation for understanding many, many of the most important processes and phenomena in nature. ◦ Differential Equations regularly appear in Physics. They appear when we have got rate of change of certain variable. ◦ A differential is an equation that have derivates of the variable. A derivative allows to find the value that a function takes at an exact value. ◦ The traditional formula gives an average value. For example the speedometer of a car may show that for the last 5 minutes you have been driving at 60 mph. ◦ However during those 5 minutes you had to sped up to pass a car, and you had to slowdown to let a car enter the highway..
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  There are lotsof application in physics, using differential equations ◦ Radioactive chains of decay ◦ The differential equation for the number N of radioactive Nuclei, which have not yet decayed is well known from elementary high school. dN/dt =-kN ◦ Linear motion equations For a motion along the x axis, we have the well known concepts. Velocity: V=dx/dt , acceleration: a=dv/dt = d2x/dt2 and Newton’s 2. law: F=ma= m*dv/dt = m d2x/dt2
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  ◦ The motionof a projectile ◦ This is also a motion equation, but in a 2 dimensional way. ◦ So, here also we use differential equations to solve the problems. ◦ Damped harmonic oscillation ◦ A harmonic oscillation is a linear movement (along an axis), where the resulting force is always directed against and proportional to the distance to the position of equilibrium. If the motion is along the x – axis, then the equation of motion is: ◦ F = -kx ◦ ma = -kx ◦ a = - k/m * x ◦ a = - d2x/dt2 = -ω2x ◦ Forced harmonic oscillations without damping ◦ F = - kx + Fext ◦ m d2x/dt2 = - kx + Fext
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  ◦ Since derivativescan provide the value of a function at an exact value, therefore using differential allows more accurate result, which is the ideal answer in the solution of a problem ( besides many problems that can be solve only using differential equation) ◦ Examples of insight provided by solution to the applicable differential equation are: flow of fluids, stresses on structures, material change by chemical process and even the interaction of atoms and molecules at the microscopic level. ◦ Technological progress in very many scientific disciplines would not have occurred without solution to differential equations to guide us and provide confident solution and insight. ◦ Scientists love it when quantity y varies proportionally to quantity x, then they can say y = mx, almost as good is when a constant offset is involved and they can say y = mx + c (the equation of a straight line). ◦ Unfortunately, things in real life usually vary non-linearly and y = aekx| a sin(x) | a cos(x) | a tan(x) or some power function in x; very often the variable x is replaced with the variable t for functions that vary with time. ◦ These relationships can be determined by differential equations: acceleration, growth, decay, oscillation, current through a diode or transistor and so on. ◦ So, almost everything in physics behaves in a non-linear fashion and requires differential equations to describe it.
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  Application of Derivatives in Biology
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  Bacterial Growth problem ◦Q. In a culture, bacteria increases at the rate proportional to the number of bacteria present. If there are 400 bacteria initially and are doubled in 3 hours , find the number of bacteria present 7 hours later ?
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  Ans : Letx be the number of bacteria , and the rate is dx/dt since the number of bacteria is proportional to the rate ,so dx/dt x if k (k>0) is the proportionality constant then, dx/dt = kx Separating the variables , we have dx/x=k.dt Since there are 400 bacteria initially and they are doubled in 3 hours, we integrate the left side of equation (1) from 400 to 800 and integrate its right side from 0 – 3 to find the value of k as follows. 1
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  ◦ Putting thevalue of k in (1) we have dx/x = (1/3 ln2) dt Next ,to find the number of bacteria present 7 hours later, we integrate the left side of (2) from 400 to x and its right side from 0 to 7 as follows. Thus there are 2016 bacteria after 7 hours. 2
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