If a continuous function f(x) satisfy the relation ₀^x t f(x-t) d t= ₀^x t f(t) d t+ x+ x-x-1 , for all real number x , then which of the following does not hold good

Step-by-step solution

Given equation can be written as ₀^x x f(t) d t- ₀^x t f(t) d t= ₀^x f(t) d t+ x+ x-x-1 differentiating x f(x)+ ₀^x f(t) d t-x f(x)=f(x)+ x- x-1 again differentiating f(x)=f^ (x)- x- x which is linear differential equation of 1^ st order whose solution gives aligned & y e^ -x =-e^ -x x+c & x=0, y=0 c=1 & f(x)=e^x- x aligned