{"version":"1.0","provider_name":"SOUL OF MATHEMATICS","provider_url":"https:\/\/soulofmathematics.com","author_name":"Rajarshi Dey","author_url":"https:\/\/soulofmathematics.com\/index.php\/author\/rajarshidey1729gmail-com\/","title":"LAGRANGE'S EQUATION - SOUL OF MATHEMATICS","type":"rich","width":600,"height":338,"html":"<blockquote class=\"wp-embedded-content\" data-secret=\"RO1b41pqNX\"><a href=\"https:\/\/soulofmathematics.com\/index.php\/lagranges-equation\/\">LAGRANGE&#8217;S EQUATION<\/a><\/blockquote><iframe sandbox=\"allow-scripts\" security=\"restricted\" src=\"https:\/\/soulofmathematics.com\/index.php\/lagranges-equation\/embed\/#?secret=RO1b41pqNX\" width=\"600\" height=\"338\" title=\"&#8220;LAGRANGE&#8217;S EQUATION&#8221; &#8212; SOUL OF MATHEMATICS\" data-secret=\"RO1b41pqNX\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\" class=\"wp-embedded-content\"><\/iframe><script type=\"text\/javascript\">\n\/* <![CDATA[ *\/\n\/*! 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Joseph-Louis Lagrange Partial differential equations can be formed by the elimination of arbitrary constants or arbitrary functions. If we have f (x, y) then we have the following representation of partial derivatives, Let F (x,y,z,p,q) = 0 be the first order differential equation. It contains three types of variables, where x and y are independent variables and z is dependent variable. A short classification of partial differential equations (PDE) &#8211; Linear equation. A first order equation f (x, y, z, p, q) = 0 is known as linear if it is linear in p, q and z, that is, if given equation is of the form P(x, y) p + Q(x, y) q = R(x, y) z + S(x, y).For example, yx2p + xy2q = xyz + x2y3 and p + q = z + xy are both first order linear partial differential equations.Semi-linear equation. A first order partial differential equation f (x, y, z, p, q) = 0 is known as a semi-linear equation, if it is linear in p and q and the coefficients of p and q are functions of x and y only i.e. if the given equation is of the form P(x, y) p + Q(x, y) q = R(x, y, z)For example, xyp + x2yq = x2y2z2 and yp + xq = (x2z2\/y2) are both first order semi-linear partial differential equations.Quasi-linear equation. A first order partial differential equation f(x, y, z, p, q) = 0 is known as quasi-linear equation, if it is linear in p and q, i.e., if the given equation is of the form P(x, y, z) p + Q(x, y, z) q = R(x, y, z)For example, x2zp + y2zp = xy and (x2 \u2013 yz) p + (y2 \u2013 zx) q = z2 \u2013 xy are first order quasi-linear partial differential equations.Non-linear equation. A first order partial differential equation f(x, y, z, p, q) = 0 which does not come under the above three types, in known as a non-liner equation.For example, p2 + q2 = 1, p q = z and x2 p2 + y2 q2 = z2 are all non-linear partial differential equations. THE EQUATION A particular Quasi-linear partial differential equation of order one is of the form Pp + Qq = R, where P, Q and R are functions of x, y, z. Such a partial differential equation is known as Lagrange equation.For Example xyp + yzq = zx is a Lagrange equation. Theorem. The general solution of Lagrange equation Pp + Qq = R, is where \u0424 is an arbitrary function and u(x, y, z) = c1 and v(x, y, z) = c2 are two independent solutions of (dx)\/P = (dy)\/Q = (dz)\/R. Here, c1 and c2 are arbitrary constants and at least one of u, v must contain z. Proof. Lets number the equations for simplification. Pp + Qq = R \u0424(u, v) = 0 u(x, y, z) = c1 and v(x, y, z) = c2 (dx)\/P = (dy)\/Q = (dz)\/R Differentiating (2) partially w.r.t. \u2018x\u2019 and \u2018y\u2019, we get equations 5 and 6, Eliminating \u2202\u0424 \/ \u2202u and \u2202\u0424\/ \u2202v between (5) and (6), we have, Hence, (2) is a solution of this equation. Taking the differentials of u(x, y, z) = c1 and v(x, y, z) = c2, we get, As u and v are independent functions, the ratios dx : dy : dz, gives Comparing equations we obtain, We can imply that, Substituting these values we get, k(Pp + Qq) = kR or Pp + Qq = R, which is the given equation (1). Therefore, if u(x, y, z) = c1 and v(x, y, z) = c2 are two independent solutions of the system of differential equations (dx)\/P = (dy)\/Q = (dz)\/R, then \u0424(u, v) = 0 is a solution of Pp + Qq = R, \u0424 being an arbitrary function. Equations (4) are called Lagrange\u2019s auxillary (or subsidiary) equations for (1). Steps for solving Pp + Qq = R by Lagrange\u2019s method. Step 1. Put the given linear partial differential equation of the first order in the standard formPp + Qq = R. \u2026(1)Step 2. Write down Lagrange\u2019s auxiliary equations for (1) namely,(dx)\/P = (dy)\/Q = (dz)\/R \u2026(2)Step 3. Solve equation (2). Let u(x, y, z) = c1 and v(x, y, z) = c2 be two independent solutions of (2).Step 4. The general solution (or integral) of (1) is then written in one of the following three equivalent forms :\u0424(u, v) = 0, u = \u0424(v) or v = \u0424(u), \u0424 being an arbitrary function. FOR TUTORIAL SHEETS AND REFINED MATERIALS SUBSCRIBE TO RECEIVE THEM VIA EMAIL. IMAGE COURTESY"}