1.0SOUL OF MATHEMATICShttps://soulofmathematics.comRajarshi Deyhttps://soulofmathematics.com/index.php/author/rajarshidey1729gmail-com/rich600338<blockquote class="wp-embedded-content" data-secret="KUWPkLVF8U"><a href="https://soulofmathematics.com/index.php/lagranges-equation/">LAGRANGE’S EQUATION</a></blockquote><iframe sandbox="allow-scripts" security="restricted" src="https://soulofmathematics.com/index.php/lagranges-equation/embed/#?secret=KUWPkLVF8U" width="600" height="338" title="“LAGRANGE’S EQUATION” — SOUL OF MATHEMATICS" data-secret="KUWPkLVF8U" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" class="wp-embedded-content"></iframe><script type="text/javascript"> /* <![CDATA[ */ /*! This file is auto-generated */ !function(d,l){"use strict";l.querySelector&&d.addEventListener&&"undefined"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!/[^a-zA-Z0-9]/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret="'+t.secret+'"]'),o=l.querySelectorAll('blockquote[data-secret="'+t.secret+'"]'),c=new RegExp("^https?:$","i"),i=0;i<o.length;i++)o[i].style.display="none";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute("style"),"height"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):"link"===t.message&&(r=new URL(s.getAttribute("src")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener("message",d.wp.receiveEmbedMessage,!1),l.addEventListener("DOMContentLoaded",function(){for(var e,t,s=l.querySelectorAll("iframe.wp-embedded-content"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute("data-secret"))||(t=Math.random().toString(36).substring(2,12),e.src+="#?secret="+t,e.setAttribute("data-secret",t)),e.contentWindow.postMessage({message:"ready",secret:t},"*")},!1)))}(window,document); /* ]]> */ </script> https://i0.wp.com/soulofmathematics.com/wp-content/uploads/2020/09/Heat.gif?fit=960%2C763&ssl=1960763As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection. 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) – 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 – yz) p + (y2 – zx) q = z2 – 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 Ф 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 Ф(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. ‘x’ and ‘y’, we get equations 5 and 6, Eliminating ∂Ф / ∂u and ∂Ф/ ∂v 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 Ф(u, v) = 0 is a solution of Pp + Qq = R, Ф being an arbitrary function. Equations (4) are called Lagrange’s auxillary (or subsidiary) equations for (1). Steps for solving Pp + Qq = R by Lagrange’s method. Step 1. Put the given linear partial differential equation of the first order in the standard formPp + Qq = R. …(1)Step 2. Write down Lagrange’s auxiliary equations for (1) namely,(dx)/P = (dy)/Q = (dz)/R …(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 :Ф(u, v) = 0, u = Ф(v) or v = Ф(u), Ф being an arbitrary function. FOR TUTORIAL SHEETS AND REFINED MATERIALS SUBSCRIBE TO RECEIVE THEM VIA EMAIL. IMAGE COURTESY