Editing Separation of variables (section)

=== Example: mixed derivatives ===

For some equations involving mixed derivatives, the equation does not separate as easily as the heat equation did in the first example above, but nonetheless separation of variables may still be applied. Consider the two-dimensional [[biharmonic equation]]

:<math>\frac{\partial^4 u}{\partial x^4} + 2\frac{\partial^4 u}{\partial x^2\partial y^2} + \frac{\partial^4 u}{\partial y^4} = 0.</math>

Proceeding in the usual manner, we look for solutions of the form

:<math>u(x,y) = X(x)Y(y)</math>

and we obtain the equation

:<math>\frac{X^{(4)}(x)}{X(x)} + 2\frac{X''(x)}{X(x)}\frac{Y''(y)}{Y(y)} + \frac{Y^{(4)}(y)}{Y(y)} = 0.</math>

Writing this equation in the form

:<math>E(x) + F(x)G(y) + H(y) = 0,</math>

Taking the derivative of this expression with respect to <math> x </math> gives <math> E'(x)+F'(x)G(y)=0 </math> which means <math> G(y)=const. </math> or <math> F'(x)=0 </math> and likewise, taking derivative with respect to <math> y </math> leads to <math> F(x)G'(y)+H'(y)=0 </math> and thus <math> F(x)=const. </math> or <math> G'(y)=0 </math>, hence either ''F''(''x'') or ''G''(''y'') must be a constant, say −λ. This further implies that either <math>-E(x)=F(x)G(y)+H(y)</math> or <math>-H(y)=E(x)+F(x)G(y)</math> are constant. Returning to the equation for ''X'' and ''Y'', we have two cases

:<math>\begin{align}
    X''(x) &= -\lambda_1X(x) \\
    X^{(4)}(x) &= \mu_1X(x) \\
    Y^{(4)}(y) - 2\lambda_1Y''(y) &= -\mu_1Y(y)
\end{align}</math>

and

:<math>\begin{align}
    Y''(y) &= -\lambda_2Y(y) \\
    Y^{(4)}(y) &= \mu_2Y(y) \\
    X^{(4)}(x) - 2\lambda_2X''(x) &= -\mu_2X(x)
\end{align}</math>

which can each be solved by considering the separate cases for <math>\lambda_i<0, \lambda_i=0, \lambda_i>0</math> and noting that <math>\mu_i=\lambda_i^2</math>.