更改

If  is a linear function of  and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.

If  is a linear function of  and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.

如果{{mvar|f}}是函数及其导数的线性函数，则偏微分方程称为线性函数。线性偏微分方程的常见例子包括热方程、波动方程、拉普拉斯方程、亥姆霍兹方程方程、克莱因-高登方程和泊松方程。

如果{{mvar|f}}是函数{{math|''u''}}及其导数的线性函数，则偏微分方程称为线性函数。线性偏微分方程的常见例子包括热方程、波动方程、拉普拉斯方程、亥姆霍兹方程方程、克莱因-高登方程和泊松方程。

A relatively simple PDE is

A relatively simple PDE is

This relation implies that the function  is independent of . However, the equation gives no information on the function's dependence on the variable . Hence the general solution of this equation is

This relation implies that the function  is independent of . However, the equation gives no information on the function's dependence on the variable . Hence the general solution of this equation is

这个关系意味着函数是独立于{{mvar|x}}的。然而，这个方程没有给出关于函数和自变量的相关性的信息。因此，这个方程的通解是

这意味着函数{{math|''u''(''x'',''y'')}}独立于{{mvar|x}}的。然而，这个方程没有给出关于函数和自变量的相关性的信息。因此，这个方程的通解是

where  is an arbitrary function of . The analogous ordinary differential equation is

where  is an arbitrary function of . The analogous ordinary differential equation is

其中{{mvar|f}}是{{mvar|y}}的任意函数。类似的常微分方程是

其中{{mvar|f}}是{{mvar|y}}的任意函数。

 

where  is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions.

where  is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions.

这里的{{mvar|c}}是常量。这两个例子说明常微分方程的一般解包含任意常数，但偏微分方程的解包含任意函数。

这里，{{mvar|c}}是一个任意常量。