Mathquery

Non_ Linear Partial Differential Equation Of First Order :          Let us consider a differential equation of the form F(x,y,p,q) = 0 in which the function F is not necessarily linear in p and q .   Singular solution / Singular Integral :               The equation of the envelop of the surface represented by the complete integral of a partial differential equation is called its singular solution or singular integral .       The envelop of the surface F(x,y,a,b)=0  is obtained by eliminating a and b from the equations F = 0 , ∂F/∂a = 0,  ∂F/∂b = 0  Charpit's Method :          Let us consider the equation f(x,y,p,q) = 0 then Charpit's subsidiary equations are given by   dx/ [∂f/∂p] = dy/ [∂f/∂q]                      = dz/[p(∂f/∂p) + q(∂f/∂q)]     ...

Complete Solution Or Complete Integral Of Partial Differential Equations :   Definition :              A solution of partial differential equation is said to be a complete solution or complete integral if it contains as many arbitrary constants as there are independent variables . Definition Of General Solution Or Integral :        A general solution or integral of a partial differential equation is a relation involving arbitrary functions which provides a solution to that equation . Linear Partial Differential Equation Of The First Order :        A partial differential equation of first order is said to be linear if it is of the first degree in P and Q otherwise it is            non linear . Example :   (i) Linear Partial Differential                    ...

Intervals Other Than [-π,π] :             So far we have considered the interval [-π,π] only . It was just a matter of convenience , otherwise any finite interval could have been used . We now show that by effecting certain transformations , any finite interval can be made to correspond to the interval      [-π,π] . The Interval [0,2π] :               If f is bounded , integrable and piecewise monotonic in [0,2π] , then the sum of the series                         ∞          1/2 a₀ + Σ (aₙ cos nx + bₙ sin nx)                       n=1                             2π  where aₙ = 1/π ∫ f cos nx dx ,                  ...

Half Range Series :         With the help of the Main Theorem and those of even and odd functions , we now consider the expansion of a function over the interval [0,π] in terms of (i) sine terms only , (ii) cosine terms only . (i) The Sine Series :               If a function f is bounded , integrable and piecewise monotonic in [0,π] , then the sum of the sine series                                                         π     Σ bₙ sin nx , where bₙ = 2/π ∫f sin nx dx                                                       0 is equal to , 1/2 [f(x-) + f(x+)] at every point x between 0 and π , and is equal to 0 , when x=0 ,π ...