Deﬁnitions and examples The wave equation The heat equation The one-dimensional wave equation Separation of variables The two-dimensional wave equation Solution by separation of variables We look for a solutionu(x,t)intheformu(x,t)=F(x)G(t). Substitution into the one-dimensional wave equation gives 1
Solution of Laplace’s equation (Two dimensional heat equation) The Laplace equation is. Let u = X(x) . Y(y) be the solution of (1), where „X‟ is a function of „x‟ alone and „Y‟ is a function of „y‟ alone. Now the left side of (2) is a function of „x‟ alone and the right side is a function of „t‟ alone.
May 30, 2020 · Differential Equations > Separation of Variables. Separation of Variables is a standard method of solving differential equations. The goal is to rewrite the differential equation so that all terms containing one variable (e.g. “x”) appear on one side of the equation, while all terms containing the other variable (e.g. “y”) appear on the opposite side.