Third Order Linear Homogeneous Differential Equations in Normal Form

Abstract

Let a third order linear differential equation be given in the form $$u''' + 3{p_1}u'' + 3{p_2}u' + {p_3}u = 0, $$ where p1= p1(x), p2=p2(x), p3=p3(x) are functions defined on an interval (a, b), −∞≦a, b≦∞, and let p 1 ″ , p 2 ′ , p3 be continuous functions of x∈(a, b), ′ denoting the derivative of a function with respect to the independent variable. Let x0∈(a, b). By the transformation \( u = y{\text{ }}\exp {\text{ }}\left( { - \int_{x0}^x {{p_1}\left( t \right)dt} } \right) \) , the above differential equation takes the form $$y'''{ + _3}{P_2}y' + {P_3}y = 0, $$ where P3=p3−3p1p2+2p 1 3 −p 1 ″ , P2=p2−p 1 2 −p 1 ′ . By writing $$ A = \frac{3}{2}{P_2},{\text{ }}b = {P_3} - \frac{3}{2}{P'_2}, $$ we obtain the following differential equation for y, $$ y''' + 2Ay' + (A' + b)y = 0. $$ (a)