IGNOU MPH 1 2026 - Use the method of separation of variables to reduce the Laplace's

Question

Use the method of separation of variables to reduce the Laplace's equation $\nabla^2 f = 0$ into three ODEs.

19 Jan 2026

Answer :

Word Count : 256

We start with Laplace's equation in three dimensions: [ \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = 0 ] Assume a solution of the form [ f(x,y,z) = X(x) Y(y) Z(z) ] where (X, Y, Z) are functions of (x, y, z) respectively. Substituting into the Laplace equation: [ \frac{\partial^2}{\partial x^2} (X Y Z) + \frac{\partial^2}{\partial y^2} (X Y Z) + \frac{\partial^2}{\partial z^2} (X Y Z) = 0 ] Since (Y) and (Z) are independent

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