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If y = f (x) is the solution of difieren...

If `y = f (x)` is the solution of difierential equation.` e ^(y) ((dy)/(dx )-2)=e ^(3x)` such that `f(0) =0,` then f (2) is equal to :

A

3

B

5

C

6

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
Put `e ^(y) =t implies e ^(y) dy = dt implies (dt)/(dx) - 2t =e ^(3x )`
`L.F. = e int ^(-2dx ) =e ^(-2x)`
`t. e ^(-2x) = int 2 ^(3x ) . E ^(-2x) dx`
`t. e ^(-2x) = int e ^(x) dx = e ^(x) + x, " "e ^(y) e ^(-2x)= e ^(x) +c`
Put `x =0, y =0` we get `e ^(0) . e ^(0) =1 +c`
` implies e ^(y) e ^(-2x ) = e ^(x)`
` e ^(y) =e ^(3x ) implies y =3x implies f(x) =3x`
`f (2) =6`
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