Showing posts from December, 2013

Motion Of A Particle

Principles in  the formation of the Equation of Motion of a Particle :-  

For Motion in Straight Line ;

Let the mass of the body (particle) be    m    and let the distance of the particle measured from a suitable origin be   x   at the time  t   .

Then acceleration is      (d/dt)(dx/dt)

therefore by Second Law of Motion

m[(d/dt)(dx/dt)] = Forces in the direction of    x   increasing

The forces usually are ;-

1)---   g    vertically downwards in a gravitational field like earth .
2)---  tension in a string
3)---  reactions or stress at point where the point may be in contact with other particles .
4)---  forces of attraction 
5)---  forces of resistance to motion by any means may be by atmosphere , friction, winds or by any other means .

In  taking the forces we must fix the sign properly if the forces acts along the line .

If a force  F   acts at an angle    A     to the straight line , then the resolved part of the force along the line is    FcosA  . If   acts on on a particle an…

Equations of First Order

In this post we will discuss about the equation of first order but not first degree .It is usually denoted

dy/dx    by   p .

Thee are three types of such equations

1) Equations solvable for .
2)Equation solvable for    y  .
3)Equation solvable for    .

Equation Solvable for   p   :- 

examples like this    p.p +2py  cot x = y.y   and its solution is

[y-(c/1+cos x)][y-(c/1-cos x)] = 0

is the equation solvable for   p   .

Equation Solvable for    y   :-

Let in the given differential equation , on solving for   y   , given that ;

  y=f(x,p)  ---------------(1)

Differentiating with respect to    , we obtain ;


so that we obtain a new differential equation with variables     and     .

Suppose that it is possible to solve the equation

Let the solution be
F(x,p,c)=0  ----------(2)
                          where     is the arbitrary constant .

The equation of  (1)  may be exhibited in either of the two forms . We may either eliminate   p   betwee…

Equations of Tangent and Normal

Explicit Cartesian Equations :-

If      be the angle which the tangent at any point   (x, y)   on the curve     y = f (x)     makes with    x   axis then ;
  tan A = dy/dx = f' (x) 

Therefore , the equation of the tangent at any point     (x , y)     on the curve    y = f (x)      is

  Y - y = f' (x) (X - x)  -------------(1)

where   X , Y   are the current co-ordinates of any point on the tangent .

The normal to the curve    y = f (x)    at any point    (x , y)     is the straight line which  passes through that point ans is perpendicular to the tangent to the curve at the point so that its slope is ;

  -1/f (x)

Hence the equation of the normal at   (x , y)    to the curve    y= f (x)    is ;

    (X - x) + f' (x) (Y - y) = 0 

Implicit Cartesian Equations :-

If any point    (x , y) , then the curve   f (x, y) = 0

          Where   Dy/Dx    is not equivalent to  0   .

     dy/dx =  - (Df/Dx) / (Df/Dy)

Hence the equations of the tangent and the normal at any point
(x , y)   on t…

Maxima and Minima

 Greatest  and Least Value :-

In this section we shall be concerned with the application of Calculus to determining the values of a function which are greatest or least in their immediate neighborhood technically known as  Maximum and Minimum Values .

It will be assumed that    f (x)   possesses continuous derivatives of every order that come in equation .

Maximum Value of a Function :-

Let     be any interior points of the interval of definition of a function   f (x)    , if it is the greatest of all its value for values of   x   lying in some neighborhood of   c   . To b more definite and to avoid the vague words   "Some Neighborhood" we say that    f (c)    is a maximum value of function if there exists some interval     ( c - D , c + D )       around     c   such that   .

  f (c) > f (x) 

for all values of     other then      c   lying in the interval

So that ,      f (c)    is  maximum value of      f (x)     if .

f (c) > f ( c+ h )     i.e.       f ( c + h ) …

Velocity and Acceleration

Velocity and Acceleration of a Moving Particles :-

If a particle is moving along a straight line , and if at any instant     t     the position        of the particle be given by the distance   s    measured along the path from a suitable fixed point    on it , then      denoting the velocity and   f   the acceleration of the particle at the instant ,

     We have;

  v = rate of displacement

                    = rate of change of   s   with respect to time

                    = ds/dt ;

and ,           f = rate of change of velocity with respect to time

                   = dv/dt

                   = ( d/dt ) ( ds/dt )

If instead of moving in a straight line , the particle be moving in any manner in a plane , the position of a particle at any instant   t   being given by the Cartesian Co-ordinates    x , y     referred to a fixed  set of axes the components of velocity and acceleration parallel to those axes will similarly be given by

   vx = rate of displacement parallel to    …

Differential Equations

In this post we will discuss about "Formation of Ordinary Differential Equations"

Let ;     f ( x , y , c1 ) = 0  ---------------------(1)

be an equation containing   x ,  y  an  on arbitrary constant   c1   .

Differentiating  (1)  we get ;

   ( Df/Dx ) + ( Df/Dx ) (dy/dx ) = 0 ------(2) 

equation  (2)  will in general contains   c1   . If   c1  be eliminated between  (1)  and  (2)  , we shall get a relation involving   x,  y  and   dy/dx  which will evidently be a differential equation  of the first order .

Similarly , if we have an equation

  f ( x , y , c1 , c2 ) = 0 ----------------(3)

containing two arbitrary constant  c1  and  c2  , then by differentiating this twice , we shall get two equations . Now between these two equations and given equations , in all three equations , if the two arbitrary constant   c1  and   c2  be eliminated , we shall evidently get a differential equation  of the second order .

in general , if we have an equation ;

  f ( x , y , c1 , c2 ,  .....…

Homogeneous Equations

We have to understand about Homogeneous Equation in Calculus .

If  and  N  of the equation   Mdx + Ndy = 0   are both of the same degree in  x  and  y   , and are homogeneous , the equation is said to be homogeneous  . Such an equation can be put in the form

  dy/dx = f ( y/x )

Every homogeneous equation of the above type can be easily solved by putting     y = vx       
where   v  is a function of , and consequently

dy/dx = v + x ( dv/dx )  

whereby it reduced to the form     v + xdv/dx = f ( v ) 

                                      i.e.        dx/x = dv/ [ f(v) - v ] 

in which the variables are separated

A Special Form :-

The equation of the form

dy/dx = [ ( a1x +b1y +c1 ) / ( a2x + b2y +c2 ) ] 

                         where ,  [ a1/a2 is not equal to b1/b2 ] --------------(1)

can be easily solved by putting
x = x' + h 
                           and              y = y' + k    
                           where   and are constant

So that ,              dx = dx&#…

Clairaut's Equation

An equation of the form

y = px + f (p) , where p = dy/dx

is called Clairaut's Equation .

Differentiating both sides of the equation with respect to   x   , we have  ,

p = p + xdp/dx + f' (p) dp/dx  or   dp/dx {x + f' (p)} = 0

therefore , either ,  dp/dx = 0 ------------ (1)

                 or ,     x + f' (p) =0 ------------(2)

                from (1) ,    p = C --------------(3)

Now if    p    be eliminated between  (3)  and the original equation , we get   y = Cx + f(C)  as the general or complete solution of the equation .

Again , if  p  be eliminated between  (2)  an the original equation , we shall obtain a relation between   x  and  y  which also satisfies the differential equation , and as such can be called a solution  of the given equation . Since this solution does not contain any arbitrary constant nor can it be derived from the complete solution by giving any particular value to the arbitrary constant , it is called the Singular Solution of the different…

Method of Isoclines

It is only in the limited number of cases that a differential equation may be solve analytically be the preceding methods , and in many practical cases where the solution of a differential equation is needed under given initial conditions and the above methods  fail , a graphical method , the method of isoclines is sometimes adopted . We proceed to explain below this method in case of simple differential equation of the first order .

Let us consider an equation of the type

dy/dx = f (x ,y ) ------------(1)

As already explained before , the general solution of this equation involves one arbitrary constant of integration . and hence represents a family of curves and in general , one member of the family passes through a given point  ( x , y ) .

Now if in  (1)  we replace   dy/dx  by   m   we get an equation   f( x , y) = m    , which for any particular numerical value of   m  represents a curve , at every point of which the value of  dy/dx  i.e.  the slope of the tangent line to the fam…

Simpson's Rule

Approximate evaluation of a definite integral :
Simpson's Rule .

In many cases , a definite integral can not be obtained either because the quantity to be integrated can not be expressed as a mathematical function , or because the indefinite integral of the unction itself can not be determined directly . In such cases formula of approximation are used . One such important formula is Simpson's Rule . By this rule the definite integral of any function is expressed in terms of the individual values of any number of ordinates within the interval , by assuming that the function within each of the small ranges into which the whole interval may be divided can be represented to a sufficient degree of approximation by a parabolic function .

In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:
Simpson's rule also corresponds to the three-point Newton-Cotes q…

Areas of Plane Curves

Areas in Cartesian Co - ordinate :-

Suppose we want to determine the area A1 bounded by the curve   y = f(x)   , the x - axis an two fixed ordinates   x = a   and   x = b    . The function   f(x)  , is supposed to be single - valued , finite and continuous in the interval  (a , b ) .

The process of finding the area , bounded by any defined contour line is called Quadrature , "the term meaning the investigation of the size of a square which shall have the same area as that  of the region under consideration" .

Here are the example of some Plain Curves :-

NameImplicit equationParametric equationAs a functiongraphStraight lineCircle

Standared Methods of Integration

The different methods of Integration will aim at reducing a given Integral to one of the Fundamental or known Integrals . As a matter of facts , there are two principal processes :

1)   The method of substitution , i.e. the change of the independent variable .

Integration by substitution can be derived from the fundamental theorem of calculus as follows. Let ƒ and ϕ be two functions satisfying the above hypothesis that ƒ is continuous on I and ϕ′ is continuous on the closed interval [a,b]. Then the function ƒ(ϕ(t))ϕ′(t) is also continuous on [a,b]. Hence the integrals
in fact exist, and it remains to show that they are equal.
Since ƒ is continuous, it possesses an antiderivative F. The composite function Fϕ is then defined. Since F and ϕ are differentiable, the chain rule gives
Applying the fundamental theorem of calculus twice gives
which is the substitution rule.

2) Integration by Parts ;

Integrating the product rule for three multiplied functions, u(x), v(x), w(x), gives a simi…