Putnam 1983 B3

Assume that the differential equation $$y'''+p(x)y''+q(x)y'+r(x)y=0$$ has solutions $y_1(x)$ , $y_2(x)$ , $y_3(x)$ on the real line such that $$y_1(x)^2+y_2(x)^2+y_3(x)^2=1$$ for all real $x$ . Let $$f(x)=y_1'(x)^2+y_2'(x)^2+y_3'(x)^2.$$ Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation $$y'+Ap(x)y=Br(x).$$