{"id":1296,"date":"2020-09-07T11:59:45","date_gmt":"2020-09-07T06:29:45","guid":{"rendered":"https:\/\/soulofmathematics.com\/?page_id=1296"},"modified":"2021-07-13T12:15:08","modified_gmt":"2021-07-13T06:45:08","slug":"lagranges-equation","status":"publish","type":"page","link":"https:\/\/soulofmathematics.com\/index.php\/lagranges-equation\/","title":{"rendered":"LAGRANGE’S EQUATION"},"content":{"rendered":"\n

As 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.<\/p>Joseph-Louis Lagrange<\/cite><\/blockquote>\n\n\n\n

Partial differential equations can be formed by the elimination of arbitrary constants or arbitrary functions. If we have f (x, y)<\/em> then we have the following representation of partial derivatives,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Let F (x,y,z,p,q)<\/em> = 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.<\/p>\n\n\n\n

A short classification of partial differential equations (PDE) –<\/p>\n\n\n\n

Linear equation<\/em><\/strong>. 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<\/em>, yx2<\/sup>p + xy2<\/sup>q = xyz + x2<\/sup>y3<\/sup> and p + q = z + xy are both first order linear partial differential equations.
Semi-linear equation<\/em><\/strong>. 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<\/em>, xyp + x2<\/sup>yq = x2<\/sup>y2<\/sup>z2<\/sup> and yp + xq = (x2<\/sup>z2<\/sup>\/y2<\/sup>) are both first order semi-linear partial differential equations.
Quasi-linear equation<\/em><\/strong>. 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<\/em>, x2<\/sup>zp + y2<\/sup>zp = xy and (x2<\/sup> \u2013 yz) p + (y2<\/sup> \u2013 zx) q = z2<\/sup> \u2013 xy are first order quasi-linear partial differential equations.
Non-linear equation<\/em><\/strong>. 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<\/em>, p2<\/sup> + q2<\/sup> = 1, p q = z and x2<\/sup> p2<\/sup> + y2<\/sup> q2<\/sup> = z2<\/sup> are all non-linear partial differential equations.<\/p>\n\n\n\n

THE EQUATION<\/h3>\n\n\n\n

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.<\/p>\n\n\n\n

Theorem<\/strong>. The general solution of Lagrange equation Pp + Qq = R, is <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

where \u0424 is an arbitrary function and u(x, y, z) = c1<\/sub> and v(x, y, z) = c2<\/sub> are two independent solutions of (dx<\/em>)\/P = (dy<\/em>)\/Q = (dz<\/em>)\/R. Here, c1<\/sub> and c2<\/sub> are arbitrary constants and at least one of u, v must contain z.<\/p>\n\n\n\n

Proof<\/strong>. Lets number the equations for simplification.<\/p>\n\n\n\n

  1. Pp + Qq = R<\/li>
  2. \u0424(u, v) = 0<\/li>
  3. u(x, y, z) = c1<\/sub> and v(x, y, z) = c2<\/sub><\/li>
  4. (dx<\/em>)\/P = (dy<\/em>)\/Q = (dz<\/em>)\/R<\/li><\/ol>\n\n\n\n

    Differentiating (2) partially w.r.t. \u2018x\u2019 and \u2018y\u2019, we get equations 5 and 6,<\/p>\n\n\n\n

    \"\"
    and<\/figcaption><\/figure>\n\n\n\n
    \"\"
    respectively.<\/figcaption><\/figure>\n\n\n\n

    Eliminating \u2202<\/strong>\u0424 \/ \u2202<\/strong>u and \u2202<\/strong>\u0424\/ \u2202<\/strong>v between (5) and (6), we have,<\/p>\n\n\n\n

    \"\"
    or,<\/figcaption><\/figure>\n\n\n\n
    \"\"
    or,<\/figcaption><\/figure>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

    Hence, (2) is a solution of this equation.<\/p>\n\n\n\n

    Taking the differentials of u(x, y, z) = c1<\/sub> and v(x, y, z) = c2<\/sub>, we get,<\/p>\n\n\n\n

    \"\"
    and,<\/figcaption><\/figure>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

    As u and v are independent functions, the ratios dx : dy : dz, gives<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    Comparing equations we obtain,<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    We can imply that,<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n
    \"\"
    and<\/figcaption><\/figure>\n\n\n\n
    \"\"<\/figure>\n\n\n\n

    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<\/sub> and v(x, y, z) = c2<\/sub> 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.<\/p>\n\n\n\n

    Equations (4) are called Lagrange\u2019s auxillary (or subsidiary) equations for (1).<\/em><\/p>\n\n\n\n

    \"\"
    First Order Quasi-linear PDE<\/figcaption><\/figure><\/div>\n\n\n\n

    Steps for solving Pp + Qq = R by Lagrange\u2019s method.<\/h4>\n\n\n\n

    Step 1<\/em><\/strong>. Put the given linear partial differential equation of the first order in the standard form
    Pp + Qq = R. \u2026(1)
    Step 2<\/em><\/strong>. Write down Lagrange\u2019s auxiliary equations for (1) namely,
    (dx)\/P = (dy)\/Q = (dz)\/R \u2026(2)
    Step 3<\/em><\/strong>. Solve equation (2). Let u(x, y, z) = c1<\/sub> and v(x, y, z) = c2<\/sub> be two independent solutions of (2).
    Step 4<\/em><\/strong>. 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.<\/p>\n\n\n\n

    FOR TUTORIAL SHEETS AND REFINED MATERIALS SUBSCRIBE TO RECEIVE THEM VIA EMAIL.<\/p>\n\n\n\n

    IMAGE COURTESY<\/a><\/p>\n\n\n\n\t

    \n\t\t\n\t\t\t

    \n\t\t\t\t\n\t\t\t<\/p>\n\t\t\t\t\t\t\t\t\t\n