Mathquery

How To Multiply Fractions With Whole Numbers  :       Multiplication of two fractional numbers or rational numbers is  solved only in three steps i.e.       Before we discuss the method of how to multiply fractions we have to understand that what is a fractional number or a rational number ?       So let us go through definition, Definition Of Fraction or Rational Numbers :               A fraction or rational number is of the form p/q, where p and q are integers and q≠0.           Then here is the question arises that what is an integer?         To answer this question let us take examples,      The set of numbers like .............-3,-2,-1,0,1,2,3..........      are called as the set of integers.        Now come to our initial question of  h...

        WELCOME TO MATHEMATICS             I n this mathematics session I shall prove that , under suitable conditions, ' the derivative of the integral and the integral of the derivative are equal ' , and consequently , ' the two repeated integrals are equal for continuous functions '.          Leibnitz's Rule In Mathematics:                If f is defined and continuous on the rectangle R = [a,b;c,d] , and if    (i)  fₓ(x,y) exists and is continuous on the rectangle R , and                      d   (ii) g(x) = ∫ f(x,y) dy , for x∈ [a,b]                     c then g is differentiable on  [a,b] and                            d   ...

Uniform Convergence :        Uniform Convergence is a part of real analysis which is discussed in detail below. Definition :             Let (X,d) be a metric space and f be a function from X to R . Also for each n ∈ ℕ let fₙ : X ---> R . Then , the sequence of functions <fₙ> converges pointwise to the function f , if for each x ∈ X , the sequence of real numbers < fₙ(x) > converges to the real number f(x) .                   Therefore <fₙ(x) > converges pointwise to f if  lim   fₙ(x) = f(x)    ∀ x ∈ X .                              n-->∞ For Example :            Let < fₙ > be the sequence defined by fₙ  : R --> R such that fₙ(x) = x / n  ∀ x∈R , n∈N  . Show that the sequence conver...

Clairaut's Differential Equations :           The equation y= px+ f(p) .  .    .  .   . (1) of first degree in x and y is called the ' Claitaut's equation ' after the name of Alex Claude CLAIRAUT(1713_1765).        Differentiating w.r.t x, bearing in mind that p= dy/ dx, we obtain          p= xdp/dx+ p+ f(p)dp/ dx or     [x+f(p)]dp/dx=0. Equating each factor to zero, we get     dp/ dx=0.      .....(2) and x+ f(p)=0.     .......(3) Integrating (2) we obtain p=c ( a constant). Putting this value of p into (1), we find its complete integral    y= cx+ f(c).       ............(4)    which represents geometrically, a family of straight lines.   Elimination of p between (3) and (1) leads to a singular solution.   Example :           ...

Exact Differential Equations :             The differential equation of the form M(x,y)dx+ N(x,y)dy=0 is called exact differential equation if δM/δy=δN/δx.         And the differential equations of the form M(x,y)dx+ N(x,y)dy=0 is called non exact differential equations if δM/δy≠δN/δx.       And it's solution is given by  ∫M(x,y)dx + ∫N(x,y)dy=c     y as.           Terms don't contain                                       x constant Let's discuss some examples related to this form, Example :      1. Solve the differential equation     ( ycosx+ siny+y)dx+(sinx+ x cosy+x)dy=0 if it is exact. Solution :             Here M= ycosx+siny+y and N=sinx+xcosy+x  Therefore δM/δy= cosx+cosy+1 and δN/δx=cosx+cos...

Homogeneous Differential Equation:             Now I want to different forms and types of differential equations .First of all we should focus on the types, that are Homogeneous Differential equations, Exact and non exact differential equations etc. The homogeneous equation must have same degree.   For example :  solve                x^2y dx - ( x^3+y^3) dy = 0 Solution :              The given equation can be written in the form          dy/ dx = x^2y/x^3+y^3 Putting y= vx, we have   v+xdv/dx= x^2.vx/x^3+v^3x^3 = v/ 1+v^3 => x dv/dx = v/1+v^3 - v                     = -v^4/1+v^3 =>(1+v ³/v⁴)dv = - dx/x =>(1/v⁴+1/v)dv = - dx/x Integrating,      -1/3v³+log(v) = - log(x)+log(c) => log(vx/c)=1/3v³ => vx/c = exp(1/3v³) =...