PROBLEM 4.1 KNOWN:Method of separation of variables for two-dimensional, steady-state conduction. FIND:Show that negative or zero values of λ2, the separation constant, result in solutions which annot satisfy the boundary conditions. cSCHEMATIC:ASSUMPTIONS:(1) Two-dimensional, steady-state conduction, (2) Constant properties. ANALYSIS:From Section 4.2, identification of the separation constant λ2leads to the two ordinary differential equations, 4.6 and 4.7, having the forms 222222d Xd YX0 Y0dxdy+=−=λλ(1,2) and the temperature distribution is ()()()x,yX xY y .=⋅θ(3) Consider now the situation when λ2= 0. From Eqs. (1), (2), and (3), find that()()()12341234XCC x, YCC y and x,yCC x CC y .=+=+=++θ(4) Evaluate the constants - C1, C2, C3and C4- by substitution of the boundary conditions: ()()()()()()()()()()()()1234123432424x0: 0,yCC0CCy0 C0y0: x,00CxCC00 C0xL: L,00CL0Cy0 C0yW: x,W00 x0CW1 ==+⋅+⋅====+⋅+⋅===+⋅+⋅===+⋅+⋅=θθθθ01==≠The last boundary condition leads to an impossibility (0 ≠1). We therefore conclude that a λ2value of zero will not result in a form of the temperature distribution which will satisfy the boundary
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Boundary value problem, ORDINARY DIFFERENTIAL EQUATIONS, Boundary conditions, C7 cos