Mathquery

Taylor's Theorem and proof: Theorem :                     If f(x,y) is a function which possesses continuous partial derivatives of order n in any domain of a point (a,b) , and the domain is large enough to contain a point (a+h,b+k) with it, then there exists a positive number 0<θ<1, such that  f(a+h,b+k) = f(a,b) + (h∂/∂x + k ∂/∂y)f(a,b)             + (1/2!)(h∂/∂x + k∂/∂y)² f(a,b) + ....+{1/(n-1)!}(h∂/∂x + k∂/∂y)ⁿ⁻¹f(a,b)+Rₙ,  where  Rₙ = (1/n!)(h∂/∂x + k∂/∂y)ⁿf(a+θh,b+θk)                                                        0<θ<1. proof: Let x = a+th , y = b+tk , where 0≤t≤1 is a parameter, and f(x,y) = f(a+th,b+tk) = φ(t) Since the partial derivatives of f(x,y) of order n are continuous in the domain under...

Bessel Equation and Bessel Function: Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number α, the order of the Bessel function.         The differential equation      x²y" + xy' + (x² - p²)y = 0 .......(1)   Where p is a real constant is called BESSEL'S(1784 _ 1846)EQUATION of order p.           It is clear that x=0 is a regular singular point of the equation . Hence we assume a solution of the form          ∞    y = Σ   cₙxⁿ⁺ʳ                           .........(2)         n=0 where c₀≠0 Substitution of the series for y , y'  and y" in equation (1) yields      ∞         ...

HERMITE Equation is another special equation like LEGENDRE Differential Equation .   It is used in the theory of linear harmonic oscillator in quantum mechanics . It is another special form of power series . The Differential Equation of the form                   y" - 2xy' + 2py = 0 ..........(1) where p is a constant , is called HERMITE DIFFERENTIAL EQUATION.       Since -2x and 2p are analytic , x= 0 is an ordinary point of equation (1) and has a power series solution  valid for all x .                     ∞ Let      y =   Σ   cₙ xⁿ           ........(2)                   n= 0 be a solution of equation (1) .    Substituting for y, y' ,y" from equation (2) in equation (1) , we obtain      ∞      ...

LEGENDRE Differential Equation: LEGENDRE  Differential Equation and Polynomial  has a great role in the area of Differential Equations , which is like a planet in the universe of Mathematics. This equation and polynomial is not only useful for solving mathematical differential equations but also used in physics.                 The Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind.        The equation            (1-x²)y" - 2xy' +p(p+1)y =0 . ......(1) Where p is a real constant is known as Legendre differential equation and it occurs in many areas of mathematics and physics . To obtain a solution of equation (1) we shall ...

Mathematics is thuniverse of knowledge. From starting of universe it is a secret of all objects of this universe. Every ending has a starting. Differential Equations is the part of  universe of Mathematics.              Differential equations are divided into two parts those are Ordinary Differential Equations and Partial Differential Equations. For example : dy/dx + xy² = x²           It is an ordinary differential equation.                            ∂u/∂t + (∂u/∂x)² = 4           It is a partial differential equation.               I have already discussed the theorems related to ordinary differential equations. The theorems and equations are all important to the solutions of ordinary differential equations.  The geometrical structure for ordinary  differential equations...

    As I have discussed previously, there are many methods and related theorems to solve a differential equation.         So now I will discuss all theorems and applications related to it.It may be longer but beneficial for solving problems of differential equations.   Theorem_1 : The differential equation           M(x,y) dx + N(x,y)dy= 0 is exact iff          ∂M/∂y= ∂N/∂x Proof :                 If the given equation is exact , we have d(u(x,y))= M(x,y) dx + N(x,y) dy. ...   (1) and we know that d(u(x,y)) = ∂u/∂x dx + ∂u/∂y dy             .......(2) Consequently by comparison of (1) and(2) ∂u/ ∂x= M(x,y) , ∂u/ ∂y = N(x,y)          .........(3) Moreover,        ∂M/∂y=∂²u/ ∂y∂x and ∂N/∂x=∂²u/ ∂x∂y and because of ∂²u/∂y∂x =∂²u/∂x∂y, we get    ...

Hello my friends,             I want discuss something about the famous equation called LAGRANGE EQUATION.  This equation is used to solve differential equations of higher degree like CLAIRAUT EQUATION.         Now ,                  The equation y= xg(p)+ f(p) is associated with the name of JOSEPH LOUIS LAGRANGE (1736_1813).             This is a generalised form of CLAIRAUT's EQUATION .   If we put g(p) =p, it is in CLAIRAUT's form.             Differentiating w.r.t x and putting dy/ dy/dx = y'= p, we have         p= g(p) + xg'(p)  dp/dx+f'(p) dp/dx   or [p-g(p)] dx/dp= xg'(p)+ f'(p)    or        dx/dp= g'(p)x/p-g(p) + f'(p)/p-g(p)..(1)   The equation (1) is a linear differential equation in x and dx/dp and it is integrable using ...