Question

设f(x)满足af(x)+bf(1-x)=c/x,a,b,c为常数,且绝对值a,b不等,求f(x)

Answer 1

af(x)+bf(1-x)=c/x, (1)

将x换为(1-x)代入得,af(1-x)+bf(1-(1-x))=c/(1-x),

即af(1-x)+bf(x)=c/(1-x), (2)

a*(1) a^2f(x)+abf(1-x)=ac/x,

b*(2) abf(1-x)+b^2f(x)=bc/(1-x),

相减,因为绝对值a,b不等,所以a^2-b^2不等于0,

得,

(a^2-b^2)f(x)=ac/x-bc/(1-x)

f(x)=c(a/x-b/(1-x))/(a^2-b^2)