MathDB

B3

Part of 1983 Putnam

Problems(1)

PDE (1+p(x))y''+y'q(x)+yr(x)=0 implies y'+Ayp(x)=Br(x)

Source: Putnam 1983 B3

9/16/2021
Assume that the differential equation y′′′+p(x)y′′+q(x)y′+r(x)y=0y'''+p(x)y''+q(x)y'+r(x)y=0has solutions y1(x)y_1(x), y2(x)y_2(x), y3(x)y_3(x) on the real line such that y1(x)2+y2(x)2+y3(x)2=1y_1(x)^2+y_2(x)^2+y_3(x)^2=1for all real xx. Let f(x)=y1′(x)2+y2′(x)2+y3′(x)2.f(x)=y_1'(x)^2+y_2'(x)^2+y_3'(x)^2.Find constants AA and BB such that f(x)f(x) is a solution to the differential equation y′+Ap(x)y=Br(x).y'+Ap(x)y=Br(x).
differential equationdifferential equationscalculus