Someone argues: "Take and on the interval

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations

Conditions of Validity, Ch Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise

Problem

Someone argues: "Take y1(x)=x3y_1(x)=x^3 and y2(x)=x2∣x∣y_2(x)=x^2|x| on the interval (−1,1)(-1,1). A short calculation shows their Wronskian W(x)=y1y2′−y1′y2W(x)=y_1y_2'-y_1'y_2 is identically zero on (−1,1)(-1,1). By the Wronskian independence test, a vanishing Wronskian means the two functions are linearly dependent. Therefore y1y_1 and y2y_2 cannot both be solutions of any second-order linear equation y′′+p(x)y′+q(x)y=0y''+p(x)y'+q(x)y=0 with p,qp,q continuous on (−1,1)(-1,1)." Find the error in this reasoning, and state whether y1y_1 and y2y_2 are in fact linearly dependent on (−1,1)(-1,1).

Related theory

Browse the problem library