Differential Equations II

In the previous post on differential equations, we looked at various differential equations of 1st order and 1st degree; but we had no clue about their nature, behavior and the fields of their application.

Since derivatives are a tool to measure the rate of change of one quantity w.r.t. other, the differential equations are used to model various physical, chemical, biological, economical, psychological activities.

  • Solving a D.E.

Since there is no universal method known, which will solve any differential equation, various techniques have been developed. These include both analytical methods as well as numerical methods. For example, in computational fluid dynamics, the numerical methods are used to obtain approximate solutions.

  • Orthogonal Trajectories

{Orthogonal \ \equiv \ Perpendicular}

In this type of problems, we are asked to find a family of curves, which would be orthogonal to given family of curves. Recall that product of slopes of 2 perpendicular lines is -1. We extend this idea to curves.

Two curves are orthogonal, when the tangents at the point of intersection of the curves are perpendicular to each other. Also, slope of tangent to the curve y=f(x) is given by {\frac {dy}{dx}}.

First, form a D.E. from the equation of family of curves. Replace \frac {dy}{dx} with  - \frac {dx}{dy} and then solve the D.E. to get the equation of family of orthogonal trajectories.

Note that

{\frac {dy}{dx} \times - \frac {dx}{dy} = -1}

The term gradient is related to orthogonal trajectories, which we will see in vector calculus.

To find the orthogonal trajectories of curves in polar coordinates, replace \frac {dr}{d \theta} by - r^2 \frac {d \theta}{dr}.

In fluid mechanics, the equipotential lines and streamlines are orthogonal to each other. In heat transfer, the direction of conduction of heat is perpendicular to the lines of constant temperature (isotherms).

  • Newton’s Law of Cooling

The law holds approximately true when the heat transfer is via. convection. It states that the rate of heat loss of the body is proportional to the difference between temperatures of the body and of the surroundings. Thus, if the surrounding temperature is \theta_o, from the law,

{\frac {d \theta}{dt} \propto (\theta - \theta_o)}


{\frac {d \theta}{dt} = k (\theta - \theta_o)}

The sign of k will change according to the direction of heat transfer. (-ve if the body is losing heat)

The problem is to find the temperature at an instant of time in the future. This is done by integrating the equation of Newton’s law with appropriate lower and upper limits.

Note : Isaac Newton published this anonymously as “Scala graduum Caloris. Calorum Descriptiones and signa” in Philosophical Transactions, 1701.

  • Electrical Circuits

There are 3 basic components of an electric circuit, where a change in voltage is possible. They are:

1) Resistance (R), voltage drop = i R, we saw this in Ohm’s law.

2) Capacitance (C), voltage drop =  \frac {1}{C} \times \int i dt

3) Inductance (L), voltage drop = L \times \frac {di}{dt}

To solve a differential equation, we use the Kirchoff’s voltage law, which states that the sum of all the voltages around a loop is equal to zero. We will study 3 types of circuits, R-C series, R-L series and L-C series.

R-L series circuit forms a linear D.E. in i and t. R-C series circuit also forms a linear D.E. in i and t. An L-C series circuit forms an equation of the variable separable type.

  • Important Integrals to Solve Problems on Circuits

{\int e^{at} sin (bt) dt = \frac {e^{at}}{a^2+b^2} [a \ sin (bt) - b \ cos (bt)] + c}

{\int e^{at} cos (bt) dt = \frac {e^{at}}{a^2+b^2} [a \ cos (bt) + b \ sin (bt)] + c}

This integrals are useful, when the voltage source is an alternator, E is of the form E_o sin (\omega t) or E_o cos (\omega t).

  • Rectilinear Motion

Analysis from Physics perspective – Click I and II.

The motion of a body in a straight line is rectilinear motion. We use the following differential equations:

v is velocity, a is acceleration, x is displacement, F is force, p is momentum.

{v = \frac {dx}{dt}, a = \frac {dv}{dt} = v \times \frac {dv}{dx}}

From Newton’s 2nd law,

{\sum F = \frac {dp}{dt} = \frac {d (mv)}{dt}}

Note that displacement, velocity, acceleration, force, momentum are all vector quantities. Hence, consideration of their direction is extremely important.

The resisting forces, such as air resistance, friction, damping act in a direction opposite to the direction of motion.

  • Newton’s Law of Gravitation

For a particle in close proximity with the earth, the gravitational force of attraction (or ‘g‘) is inversely proportional to the square of the distance between the center of the earth and the particle. Thus,

{\frac {d^2 r}{dt^2} \propto \frac {1}{r^2}}

Note: The least velocity of projection of a particle, such that it does not come back to earth is

{v = \sqrt {2 \times g \times R}}

  • Simple Harmonic Motion

This is another application of D.E. in mechanics. The restoring force in this type of motion is directly proportional to the displacement of the body, or

{F \propto x}

{F = -kx}

{m \times \frac {d^2x}{dt^2} = - k x}

k is the constant of spring and m is the mass of body.

Negative sign indicates that displacement and acceleration have opposite directions.

This equation can be written as

{m \ddot x + kx =0}

{i.e. \ \ddot x + \frac {k}{m} x = 0}

{i.e. \ \ddot x + \omega^2 x = 0}

The solution to this differential equation is given by x = a sin (\omega t) or x = a cos (\omega t). The only difference is the starting point of the motion, i.e. the value of x at $t=0$. If it is the mean position,  x=0, use sine function and if it is one of the extremities, x =a, use cosine function.

\omega is also known as the circular frequency.

  • Spring Mass System

The spring mass system is similar to the SHM with an additional equation, mg = k \delta, where \delta is the change in length of the spring in equilibrium position. The value of \omega is obtained from this relation.

  • Heat Conduction

The law of nature is such that heat spontaneously flows from a reservoir at higher temperature to reservoir at lower temperature, until both the reservoirs attain same temperatures. Conduction is a mode of transfer of heat, the other 2 being convection and radiation. (There are separate laws for convection (Newton’s law of cooling) and radiation (Planck’s radiation law)).

The D.E. governing the rate of flow of heat is the Fourier’s law:

{\dot Q \propto A \times \frac {\partial T}{\partial x}}

\dot Q is the rate of flow of heat, A is the area through which the heat flows and \frac {\partial T}{\partial x} gives the temperature gradient in x direction. The constant of proportionality is known as the thermal conductivity, k.

{\dot Q = - k \times A \times \frac {\partial T}{\partial x}}

The applications of D.E.s in chemical engineering include Mixing of Solutions and Dissolving a Solid in a Liquid. One has to make use of the following fact to solve the problems of this kind:

rate of change of a substance inside a container = (rate of in-flow) – (rate of out-flow + rate of formation)


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