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Note for Mathematics-3 - M-3 By JNTU Heroes

  • Mathematics-3 - M-3
  • Note
  • Jawaharlal Nehru Technological University Anantapur (JNTU) College of Engineering (CEP), Pulivendula, Pulivendula, Andhra Pradesh, India - JNTUACEP
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* To eliminate 2 constants we require at least 3 equations hence we partially differentiate the above equation with respect to (w.r.t) ‘x’ and w.r.t ‘y’ to obtain 2 more equations. * From the three equations we can eliminate the constants ''a'' and ''b''. NOTE 1: If the number of arbitrary constants to be eliminated is equal to the number independent variables, elimination of constants gives a first order partial differential equation. But if the number of arbitrary constants to be eliminated is greater than the number of independent variables, then elimination of constants gives a second or higher order partial differential equation. NOTE 2: In this chapter we use the following notations: p = ∂z/∂x, q = ∂z/∂y, r = ∂2z/∂x2, s = ∂2z/(∂x∂y), t = ∂2z/∂y2 METHOD TO SOLVE PROBLEMS: Step 1: Differentiate the given question first w.r.t ‘x’ and then w.r.t ‘y’. Step 2: We know p = ∂z/∂x and q = ∂z/∂y. Step 3: Now find out a and b values in terms of p and q. Step 4: Substitute these values in the given equation. Step 5: Hence the final equation is in terms of p and q and free of arbitrary constants ''a'' and ''b'' which is the required partial differential equation. (ii) Formation of Partial Differential Equations by the elimination of arbitrary functions method: * Here it is the arbitrary function that gets eliminated instead of the arbitrary constants ''a'' and ''b''. NOTE: The elimination of 1 arbitrary function from a given partial differential equation gives a first order partial differential equation while the elimination of the 2 arbitrary functions from a given relation gives second or higher order partial differential equations. 4

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METHOD TO SOLVE PROBLEMS: Step 1: Differentiate the given question first w.r.t ‘x’ and then w.r.t ‘y’. Step 2: We know p = ∂z/∂x and q = ∂z/∂y. Step 3: Now find out the value of the differentiated function (f'' ) from both the equations separately. [(f’’) =?] Step 4: Equate the other side of the differentiated function (f'' ) which is in terms of p in one equation and q in other. Step 5: Hence the final equation is in terms of p and q and free of the arbitrary function which is the required p.d.e. * Incase there are 2 arbitrary functions involved, then do single differentiation i.e. p = ∂z/∂x, q = ∂z/∂y, then also do double differentiation i.e. r = ∂2z/∂x2, t = ∂2z/∂y2 and then eliminate (f'' ) and (f'''' ) from these equations. Worked out Examples Elimination of arbitrary constants: Ex 1: 5

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Elimination of arbitrary functions: Ex 2: ( ) ( ) Partial Differential Equations of First Order The general form of first order p.d.e is F(x, y,z,p,q)=0 .... (1) where x,y are the independent variables and z is dependent variable . Types of Solution of first order partial differential equations Complete Solution Any function f(x,y,z,a,b)=0 ........(2) involving arbitrary constants a and b satisfying p.d.e (1) is known as complete solution or complete integral of (1). General Solution 6

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Any function F (u,v)=0.......(3) satisfying p.d.e (1) is known as general solution or of (1). Linear partial differential equations of First Order Larange’s Linear Equation The equation of the form Pp+Qq=R......(1) where P,Q,R are functions of x,y,z is called Lagrange’s partial differential equation. Any function F(u,v)=0 .....(2) where u=u(x,y,z) and v=v(x,y,z) satisfying (1) is the general solution . Methods of obtaining General Solution ( ) ( ) ( ) ( ) ( ) ( ) 7

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