This tag is for questions relating to linear partial differential equations, in which the dependent variable (and its derivatives) appear in terms with degree at most one
Definition: A partial differential equation is said to be linear if the dependent variable and its partial derivatives occur only in the first degree and are not multiplied.
- A first-order PDE for an unknown function $~u(x,y)~$ is said to be linear if it can be expressed in the form $$a(x,y)~\frac{\partial}{\partial x}u(x,y)+b(x,y)~\frac{\partial}{\partial y}u(x,y)+c(x,y)~u(x,y)=d(x,y)$$
- The general linear second-order PDE in two independent variables has the form $$Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots\text{(lower order terms)}=0$$where the coefficients $~A,~ B, ~C,\cdots~$ may depend upon $~x~$ and $~y~$.
- Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables.
Examples: Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.
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