On the structure of solutions of linear partial differential equation $$\sum\limits_{i + j \leqslant n} {a_{ij} p^i q^j \varphi } = 0$$ with two independent variables and constant coefficients

1985 ◽  
Vol 6 (5) ◽  
pp. 447-462
Author(s):  
Gai Bing-zheng
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Structure Of Solutions ◽  
Partial Differential ◽  
Independent Variables ◽  
Constant Coefficients ◽  
Two Independent Variables

scholarly journals Memoir on the integration of partial differential equations of the second order in three independent variables, when an intermediary integral does not exist in general

1898 ◽  
Vol 62 (379-387) ◽  
pp. 283-285
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
General Feature ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Dependent Variables ◽  
Two Independent Variables

The general feature of most of the methods of integration of any partial differential equation is the construction of an appropriate subsidiary system and the establishment of the proper relations between integrals of this system and the solution of the original equation. Methods, which in this sense may be called complete, are possessed for partial differential equations of the first order in one dependent variable and any number of independent variables; for certain classes of equations of the first order in two independent variables and a number of dependent variables; and for equations of the second (and higher) orders in one dependent and two independent variables.

scholarly journals Different forms of Euler’s theorem forhomogeneous functions to solve partial differential equation

2021 ◽  
Vol 12 (2) ◽  
pp. 2810-2813
Author(s):  
Amit Kumar, Et. al.
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Independent Variables ◽  
Euler’S Theorem ◽  
Different Order ◽  
Two Independent Variables

In this paper we will discuss Euler’s theorem for homogenous functions to solve different order partial differential equations. We will see that how we can predict the solution of partial differential Equation using different approaches of this theorem. In fact we also consider the case when more than two independent variables will be involved in the partial differential equation whenever dependent functions will be homogenous functions. We will throw a light on one method called Ajayous rules to predict the solution of homogenous partial differential equation.

scholarly journals Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem

1959 ◽  
Vol 42 ◽  
pp. 3-5
Author(s):  
D. H. Parsons
Keyword(s):  
Differential Equation ◽  
Elementary Proof ◽  
Linear Partial Differential Equation ◽  
Partial Differential ◽  
Independent Variables ◽  
Linear Partial Differential Equations ◽  
Two Factors ◽  
Constant Coefficients ◽  
Image Position ◽  
The Right

We consider a linear partial differential equation with constant coefficients in one dependent and m independent variables, the right-hand side being zero,P being a symbolic polynomial in D1, …, Dm. If P can be decomposed into two factors, so that the equation can be writtenit is evident that the sum of any solution ofand any solution ofis also a solution of (1).

scholarly journals Nonlinear integral inequality in two independent variables

1988 ◽  
Vol 11 (1) ◽  
pp. 115-119
Author(s):  
P. T. Vaz ◽  
S. G. Deo
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Inequality ◽  
Nonlinear Partial Differential Equation ◽  
Partial Differential ◽  
Independent Variables ◽  
Nonlinear Integral Inequality ◽  
Nonlinear Inequality ◽  
Two Independent Variables

In this note, the authors obtain a generalization of the integral inequality of Bihari [1] to a nonlinear inequality in two independent variables. With the aid of this inequality a bound for the solution of a nonlinear partial differential equation is established.

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