Mathematical Methods for Physics and Engineering - Matematica.NET

15.2 LINEAR EQUATIONS WITH VARIABLE COEFFICIENTSy = x λ . This leads to an algebraic equation whose solution gives the allowed valuesof λ; the general solution is then the linear superposition of these functions.15.2.2 Exact equationsSometimes an ODE may be merely the derivative of another ODE of one orderlower. If this is the case then the ODE is called exact. The nth-order linear ODEa n (x) dn ydx n + ···+ a 1(x) dydx + a 0(x)y = f(x), (15.41)is exact if the LHS can be written as a simple derivative, i.e. ifa n (x) dn ydx n + ···+ a 0(x)y = d []b n−1 (x) dn−1 ydx dx n−1 + ···+ b 0(x)y . (15.42)It may be shown that, for (15.42) to hold, we requirea 0 (x) − a ′ 1(x)+a ′′2(x) − ···+(−1) n a n (n) (x) =0, (15.43)where the prime again denotes differentiation with respect to x. If (15.43) issatisfied then straightforward integration leads to a new equation of one orderlower. If this simpler equation can be solved then a solution to the originalequation is obtained. Of course, if the above process leads to an equation that isitself exact then the analysis can be repeated to reduce the order still further.◮Solve(1 − x 2 ) d2 y dy− 3x − y =1. (15.44)dx2 dxComparing with (15.41), we have a 2 =1− x 2 , a 1 = −3x and a 0 = −1. It is easily shownthat a 0 − a ′ 1 + a′′ 2 = 0, so (15.44) is exact and can therefore be written in the form[db 1 (x) dy ]dx dx + b 0(x)y =1. (15.45)Expanding the LHS of (15.45) we find( )d dyb 1dx dx + b d 2 y0y = b 1dx 2 +(b′ 1 + b 0 ) dydx + b′ 0y. (15.46)Comparing (15.44) and (15.46) we findb 1 =1− x 2 , b ′ 1 + b 0 = −3x, b ′ 0 = −1.These relations integrate consistently to give b 1 =1− x 2 and b 0 = −x, so (15.44) can bewritten as[d(1 − x 2 ) dy ]dx dx − xy =1. (15.47)Integrating (15.47) gives us directly the first-order linear ODEdy(dx − x)y = x + c 11 − x 2 1 − x , 2which can be solved by the method of subsection 14.2.4 and has the solutiony = c 1 sin −1 x + c 2√1 − x2− 1. ◭505