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74) we find d x = cos 8 d r - r sin 8 do, d y = sin 8 d r + r cos 8 do. 4 Solution by Transformation Upon setting Eq. 74) and Eq. 75) into Eq. 76) with A the constant of integration. In the original variables Eq. 77) which is an implicit form. (ii) A further application is that of a transformation of Legendre. 78) f ( x , Y , P) = 0. 79) Y=px-y whose differential is dY = p dx + x d p - d y = ( p - y ' ) dx + x d p = x d p . 80) dp. Thus from Eq. 81) The foregoing results transform the general first order differential equation, Eq.

82) which is likewise of first order. The solution of Eq. 82) is expressible as F ( X , Y , C ) = 0. To return to the original variables set X =p, F ( p , px - Y , C ) = 0. 84) The solution to the original problem is then obtained by eliminating p between Eqs. 78). If this is not possible the equations for x and y may be left in parametric form, with parameter p . 2 Exact Methods of Solution 40 For example the equation xp2 + 4p - 2y = 0 transforms to PX2 + 4x - 2(PX - Y) = 0 under the Legendre transformation.

Initially, at t c 0, the whole system is at temperature T,. At t = 0 the filament temperature is suddenly raised to Tf > T, by an electric current. Heat is convected to the surrounding gas and also radiated to the tube wall. The wall receives heat by convection from the gas and by radiation from the filament. Finally we assume the wall transfers heat by convection to the surrounding atmosphere at temperature T,. The dimensionless equations for this problem (see Crandall 20 1 The Origin of Nonlinear Equations Filament Fig.