Let y^ (x)+y(x) g^ (x)=g(x) g^ (x) y(0)=0, x R , where f^ (x) denotes d f(x) d x and g(x) is a given non-constant differentiable function on R with g(0)=g(2)=0 . Then, the value of y (2) is...

Step-by-step solution

d y d x +y g^ (x)=g(x) g^ (x) IF =e^ g^ (x) d x =e^ g(x) Solution is y (e^ g(x) )= g(x) g^ (x) e^ g(x) d x+C Put g(x)=t, g^ (x) d x=d t aligned & y (e^ g(x) )= t e^t d t+C = & t e^t- 1 e^t d t+C=t e^t-e^t+C & y e^ g(x) =(g(x)-1) e^ g(x) +C ( i ) aligned Given, y(0)=0, g(0)=g(2)=0 Eq. (i) becomes, aligned & y(0) e^ g(0) =(g(0)-1) e^ g(0) +C & 0=(-1) 1+C C=1 & y(x) e^ g(x) =(g(x)-1) e^ g(x) +1 & y(2) e^ g(2) =(g(2)-1) e^ g(2) +1 & where, g(2)=0 & y(2) 1=(-1) 1+1 y(2)=0 aligned