Mathquery

Special  Types  Of First Order   Differential  Equations (Part_2) : Standard Form _2 :                 Equations not involving the independent variables i.e. equation of the form                    f(z,p,q) = 0          ...........(1)        The auxiliary equations for equation(1)  are   dx/[∂f/∂p] = dy/[∂f/∂q]= dz/[p∂f/∂p+q∂f/∂q]                                  = dp/-p∂f/∂z = dq/-q∂f/∂z       ∴  dp/p = dq/q Integrating   q = ap              ............(2)   Where a is a constant .  Substituting in equation(1) , we have           f(z,p,ap) = 0  Now dz = pdx + qdy...

Special Types Of First_Order Equations  Part_1 : Standard Form 1 :                 Equations involving only p and q ,The equations of this form are                 f(p,q) = 0 .......(1) Charpit's equations take the forms     dx/[∂f/∂p] = dy/[∂f/∂q]=dz/[p∂f/∂p+q∂f/∂q]                      = dp/0 = dq/0 ∴       dp = 0 ⇒p = constant = a (say) .......(2) Substituting in equation(1) , we get                 f(a,q) = 0 ......(3) which gives q = φ(a) = constant  Hence dz = pdx + qdy                    = a dx + φ(a) dy  Which on integration yields            z = ax + φ(a)y + c .......(4) Example _ 1 : ...

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 ,                  ...