Mathquery

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 ...

Fourier Series For Even And Odd Functions : Even Function :             If f is an even function, i.e ,  f(-x) =f(x) , ∀ x , then f cos nx is an even and f sin nx is an odd function and therefore                        π        aₙ = 1/π ∫ f cos nx dx                       -π                         0                        π             = 1/π [∫ f cos nx dx + ∫ f cos nx dx]                       -π                        0                       π       ...