Mathematical Methods for Physics and Engineering - Matematica.NET

FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS(in this example equal to 3). The RHS of a homogeneous ODE can be written asa function of y/x. The equation may then be solved by making the substitutiony = vx, sothatdydx = v + x dvdx = F(v).This is now a separable equation and can be integrated directly to give∫∫dv dxF(v) − v = x . (14.19)◮Solvedydx = y ( y)x +tan .xSubstituting y = vx we obtainv + x dvdx = v +tanv.Cancelling v on both sides, rearranging and integrating gives∫∫ dxcot vdv=x =lnx + c 1.But∫∫cot vdv=cos vsin v dv =ln(sinv)+c 2,so the solution to the ODE is y = x sin −1 Ax, whereA is a constant. ◭Solution method. Check to see whether the equation is homogeneous. If so, makethe substitution y = vx, separate variables as in (14.19) and then integrate directly.Finally replace v by y/x to obtain the solution.14.2.6 Isobaric equationsAn isobaric ODE is a generalisation of the homogeneous ODE discussed in theprevious section, and is of the formdy A(x, y)=dx B(x, y) , (14.20)where the equation is dimensionally consistent if y and dy are each given a weightm relative to x and dx, i.e. if the substitution y = vx m makes it separable.476