A second order linear partial differential equation can be reduced to so-called canonical form by an appropriate change of variables ? = ?(x, y), ? = ?(x, y).

The order of an equation is defined as highest order derivative present in equation and hence from equation, \(\frac{?^2 u}{?x^2}+(\frac{?^2 u}{?x?y})^2+\frac{?^2 u}{?y^2}=x^2+y^2\), it is clear that it of 2^nd order.

Equations of form, z=px+qy+f(p,q) are known as Clairaut?s partial differential equations, named after Swiss mamatician, A. C. Clairaut (1713-1765).

The following are different standard methods used in finding solution of a differential equation:

• Variable Separable
• Homogenous Equation
• Non-homogenous Equation reducible to Homogenous Equation
• Exact Differential Equation
• Non-exact Differential Equation that can be made exact with help of integrating factors
• Linear First Order Equation
• Bernoulli?s Equation

Parabolic equations have only one family of characteristic curves. Instead of two equations like for hyperbolic equations, we have just single equation, a?_x+b?_y=0 (or a?_x+b?_y=0).

The Bessel polynomials are an orthogonal sequence of polynomials. The most accepted definition for se series was put forth by Krall and Frink (1948) as,
\(y_n (x)=?_{k=0}^n\frac{(n+k)!}{(n-k)!k!} (\frac{x}{2})^k.\)

Here we use partial differentiation.
\(u=cos?(\sqrt x+\sqrt y)\)
\(\frac{\partial u}{\partial x}= -sin?(\sqrt x+\sqrt y)�\frac{1}{2\sqrt x}\)
\(\frac{\partial z}{\partial x}=\frac{-1}{2\sqrt x}�sin?(\sqrt x+\sqrt y)\).

\(a_0 y+a_1 y’+a_2 y”+?+a_n y^{(n)}=0,\) is form which represents a homogeneous linear differential equation which has constant coefficients. In or words, it has constant coefficients if it is defined by a linear operator with constant coefficients.

Given: yzp+zxq=xy
Here, subsidiary equations are, \(\frac{dx}{yz}=\frac{dy}{zx}=\frac{dz}{xy} \)
From first two and last two terms, we get, respectively,
\(\frac{dx}{yz}=\frac{dy}{zx’}\) or xdx-ydy=0, and
\(\frac{dx}{zx}=\frac{dy}{xy’}\) or ydx-zdy=0,
Integrating se we get two solutions
x²-y²=a , y²-z²=b
Hence, general solution of given equation is,
?(x²-y², y²-z² )=0.

Equations of form Pp + Qq = R , where P, Q and R are functions of x, y, z, are known as Lagrange?s linear equation.