变量分离法是求解常微分方程和偏微分方程的一种方法。
对于一个常微分方程
 |
(1)
|
其中 
在初始值附近非零，解由下式隐式给出
 |
(2)
|
如果积分可以以闭合形式完成，并且得到的方程可以求解 
(这两个都是相当大的“如果”)，那么就得到了问题的完整解。这种技术最适用的最重要的方程是 
，指数增长和衰减方程 (Stewart 2001)。
对于函数 
和变量 
, 
, ... 的偏微分方程，可以通过进行如下形式的替换来应用变量分离法
 |
(3)
|
将得到的方程分解为一组独立的常微分方程，求解这些方程得到 
, 
, ...，然后将它们代回原始方程。
这种技术之所以有效，是因为如果独立变量函数的乘积是常数，则每个函数必须单独为常数。成功需要选择合适的坐标系，并且可能并非在所有情况下都可实现，具体取决于方程。变量分离法最早由洛必达于 1750 年使用。它在求解数学物理中出现的方程时特别有用，例如拉普拉斯方程、亥姆霍兹微分方程和薛定谔方程。
另请参阅
亥姆霍兹微分方程,
拉普拉斯方程,
偏微分方程,
Stäckel 行列式 在 MathWorld 课堂中探索此主题
此条目的部分内容由 John Renze 贡献
使用 Wolfram|Alpha 探索
参考文献
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变量分离法
引用为
Renze, John 和 Weisstein, Eric W. “变量分离法。” 来自 MathWorld——Wolfram Web 资源。 https://mathworld.net.cn/SeparationofVariables.html
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