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## Homework Statement

My textbook (

*Advanced Engineering Mathematics, seventh edition,*Kreyszig) indicates that if

*u*and

_{1}*u*are solutions to a second-order homogeneous partial differential equation, and

_{2}*c*and

_{1}*c*are constants, then

_{2}*u*where

*u = c*

_{1}u_{1}+ c_{2}u_{2}is also a solution, this is the linearity principle (which it also calls the "Fundamental Theorem", not sure why though). I am asked to prove this theorem for second-order homogeneous partial differential equations of one and two variables. I am given as a hint that the proof is very similar to the same theorem for a second order homogeneous ordinary differential equation.

## Homework Equations

First, the proof for the linearity principle for a 2nd-order homogeneous ODE of the form

*y'' + p(x)y' + q(x)y = 0.*Substitute

*y = (c*. This results ultimately in:

_{1}y_{1}+ c_{2}y_{2})*y'' + p(x)y' + q(x)y = c*

_{1}(y''_{1}+ py'_{1}+ qy_{1}) + c_{2}*(y''*_{2}+ py'_{2}+ qy_{2}) = 0I looked up also that the general form of a second-order homogeneous PDE is

*Au*where

_{xx}+Bu_{xy}+ Cu_{yy}+ Du_{x}+ Eu_{y}+ Fu + G = 0*A, B, C, D, E, F,*and

*G*are functions of

*x*and

*y.*

## The Attempt at a Solution

I tried the same approach with substitution. I let

*α*and

*β*be constants and

*u = αu*

_{1}+ βu_{2}Then

*Au*

_{xx}+Bu_{xy}+ Cu_{yy}+ Du_{x}+ Eu_{y}+ Fu + G = 0Becomes

*A(*

*αu*)_{1}+ βu_{2}_{xx}+B*(**αu*)_{1}+ βu_{2}_{xy}+ C*(**αu*)_{1}+ βu_{2}_{yy}+ D*(**αu*)_{1}+ βu_{2}_{x}+ E*(**αu*)_{1}+ βu_{2}_{y}+ F*(*+ G = 0*αu*)_{1}+ βu_{2}Finally resulting in

α(

*Au*

_{1}_{xx}+Bu_{1}_{xy}+ Cu_{1}_{yy}+ Du_{1}_{x}+ Eu_{1}_{y}+ Fu_{1}+ G) + β(*Au*_{2}_{xx}+Bu_{2}_{xy}+ Cu_{2}_{yy}+ Du_{2}_{x}+ Eu_{2}_{y}+ Fu_{2}+ G) = 0It is from here that I do not know how to continue. How do I prove from this, as in the case for homogeneous ODE's, that a linear combination of solutions is also a solution? Thank you in advance, and I seriously hope I didn't somehow wreck the formatting :)