Separation of variables
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

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

and we obtain the equation

Writing this equation in the form

Taking the derivative of this expression with respect to x {\displaystyle x} gives E ′ (x) + F ′ (x) G (y) = 0 {\displaystyle E'(x)+F'(x)G(y)=0} which means G (y) = c o n s t. {\displaystyle G(y)=const.} or F ′ (x) = 0 {\displaystyle F'(x)=0} and likewise, taking derivative with respect to y {\displaystyle y} leads to F (x) G ′ (y) + H ′ (y) = 0 {\displaystyle F(x)G'(y)+H'(y)=0} and thus F (x) = c o n s t. {\displaystyle F(x)=const.} or G ′ (y) = 0 {\displaystyle G'(y)=0}, hence either F (x) or G (y) must be a constant, say −λ. This further implies that either − E (x) = F (x) G (y) + H (y) {\displaystyle -E(x)=F(x)G(y)+H(y)} or − H (y) = E (x) + F (x) G (y) {\displaystyle -H(y)=E(x)+F(x)G(y)} are constant. Returning to the equation for X and Y, we have two cases

and

which can each be solved by considering the separate cases for λ i < 0, λ i = 0, λ i > 0 {\displaystyle \lambda _{i}<0,\lambda _{i}=0,\lambda _{i}>0} and noting that μ i = λ i 2 {\displaystyle \mu _{i}=\lambda _{i}^{2}}.
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