# LAGRANGE’S EQUATION

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.

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*, yx

^{2}p + xy

^{2}q = xyz + x

^{2}y

^{3}and p + q = z + xy are both first order linear partial differential equations.

**. 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)**

*Semi-linear equation**For example*, xyp + x

^{2}yq = x

^{2}y

^{2}z

^{2}and yp + xq = (x

^{2}z

^{2}/y

^{2}) are both first order semi-linear partial differential equations.

**. 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)**

*Quasi-linear equation**For example*, x

^{2}zp + y

^{2}zp = xy and (x

^{2}– yz) p + (y

^{2}– zx) q = z

^{2}– xy are first order quasi-linear partial differential equations.

**. 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.**

*Non-linear equation**For example*, p

^{2}+ q

^{2}= 1, p q = z and x

^{2}p

^{2}+ y

^{2}q

^{2}= z

^{2}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) = c_{1} and v(x, y, z) = c_{2} are two independent solutions of (*dx*)/P = (*dy*)/Q = (*dz*)/R. Here, c_{1} and c_{2} 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) = c
_{1}and v(x, y, z) = c_{2} - (
*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) = c_{1} and v(x, y, z) = c_{2}, 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) = c_{1} and v(x, y, z) = c_{2} 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 form

Pp + Qq = R. …(1)

**. Write down Lagrange’s auxiliary equations for (1) namely,**

*Step 2*(dx)/P = (dy)/Q = (dz)/R …(2)

**. Solve equation (2). Let u(x, y, z) = c**

*Step 3*_{1}and v(x, y, z) = c

_{2}be two independent solutions of (2).

**. The general solution (or integral) of (1) is then written in one of the following three equivalent forms :**

*Step 4*Ф(u, v) = 0, u = Ф(v) or v = Ф(u), Ф being an arbitrary function.

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