The solution of differential equation ` [ e^(-2sqrt(x))/(sqrt(x))-y/(sqrt(x))](dx)/(dy)=1, (x!=0)` is

Options

A. ` y e^(2sqrt(x)) = 2 sqrt(x)+C`
B. ` y e^(2sqrt(x)) = 3sqrt(x)+C`
C. ` 2y e^(2sqrt(x))= 3sqrt(x)+C`
D. ` y e^(sqrt(x)) = 2 sqrt(x)= +C`

Answer

d

Explanation

The Given differential equation is
` (e^(-2sqrt(x))/sqrt(x)-y/(sqrt(x)))(dx)/(dy) =1 rArr (dy)/(dx) = e^(-2sqrt(2))/(sqrt(x))=y/(sqrt(x))`
` rArr (dy)/(dx) + 1/(sqrt(x)) y = e^(-2sqrt(x))/(sqrt(x))`
On comparing with the form `(dy)/(dx) +Py = Q` , we get
` P = 1/(sqrt(x)) , Q = e^(-2sqrt(x))/(sqrt(2))`
` :. IF = e^(int 1/(sqrt(x))dx) rArr IF = e^(2x^(1//2)) = e^(2sqrt(x))`
Required solution is
`y * IF int Q xx IF dx + C`
` rArr ye^(2sqrt(x)) = int e^(2sqrt(x)) xx (e^(-2sqrt(x)))/(sqrt(2)) dx + C`
`rArr ye ^(2sqrt(x)) = int 1/(sqrt(x)) dx +C`
` :. ye ^(2sqrt(2)) = 2 sqrt(x) +C`