Mathquery

Hypergeometric Equation And Hypergeometric Function :         The linear equation      x(1-x)y" + [γ-(1+α+β)x]y'-αβy = 0, .....(1)      Where α,β and γ are constants which can take various real or complex values is called the Hypergeometric equation or the Gauss equation .     The equation (1) can be put to the form    y" + [γ-(1+α+β)x]y'/x(1-x) - αβy/x(1-x) = 0 or      y"+(1+x+x²+...)[γ-(1+α+β)x]y'/x                -αβ(1+x+x²+...)y/x = 0 or    y" + p(x)y' + q(x)y = 0,   Since xp(x) and x²q(x) can be expressed as power series , x=0 is a regular singularity . Hence the above equation can be solved by Frobenius method .  We write             ∞       y = Σ cₙ xⁿ⁺ʳ , where c₀≠0,            n=0 so that  y' = Σcₙ(n+r)xⁿ...

Special Types Of First_Order Equations Part_3 : Standard Form _ 3 :              Separable equations i.e. equations of the type                f₁(x,p) = f₂(y,q)      ............ (1)    The Charpit's auxiliary equation for equation(1) are        dx/[∂f₁/∂p] = dy/-[∂f₂/∂q]                        = dz/[p∂f₁/∂p - q∂f₂/∂q]                      = dp/-[∂f₁/∂x] = dq/-[∂f₂/∂y] ∴        dx/[∂f₁/∂p] = dp/-[∂f₁/∂x] ⇒ ∂f₁/∂x dx +∂f₁/∂p dp = 0 ⇒df₁(x,p) = 0 Integrating , f₁(x,p) = a (say) ......... (2)   Hence , equation(1) gives                    f₂(y,q) = a       .......... (3) Solving equations (2) and (3) f...

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)]     ...