1)

Let  y(x)+y(x)g(x)=g(x)g(x)y(0)=0,xϵR, where f'(x)  denoted df(x)dx 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 .....


A) 0

B) 4

C) 5

D) 6

Answer:

Option A

Explanation:

dydx+y.g(x)=g(x)g(x)

 IF= eg(x)dx=eg(x)

  solution is 

              y(eg(x))=g(x).g(x).eg(x)dx+C

Put   g(x)=t,g(x)dx=dt

 y(eg(x))=t.etdt+C

  =t.et1.etdt+C=t.etet+C

 yeg(x)=(g(x)1)eg(x)+C........(i)$

 Given , y(0)=0,g(0)=g(2)=0

  Eq.(i) becomes 

        y(0).eg(0)=(g(0)1).eg(0)+C

    0=(1).1+CC=1

   y(x).eg(x)=(g(x)1)eg(x)+1

y(2).eg(2)=(g(2)1)eg(2)+1

 where g(2)=0

  y(2).1=(1).1+1y(2)=0