Solve`[(e^(-2sqrt(x)))/(sqrt(x))-y/(sqrt(x))](dx)/(dy)=1(x!=0`

1.	Solve`[(e^(-2sqrt(x)))/(sqrt(x))-y/(sqrt(x))](dx)/(dy)=1(x!=0`
Answer» The given differential equation may be written as ` (dy )/(dx) = - (1)/(sqrtx) * y + ( e ^( -2sqrt x ))/( sqrtx )` `rArr (dy)/(dx) + (1)/(sqrt x) * y = ( e ^(-2sqrt x))/( sqrt x )" "` ... (i) This is of the form ` (dy)/(dx) + Py= Q`, where ` P = (1)/(sqrtx ) and Q = ( e ^(-2sqrtx ))/( sqrt x )` `IF = e ^(int Pdx ) = e ^(int (1)/(sqrtx) dx ) = e ^(2sqrt x ) ` So, the solution of the given differential equation is given by `y xx IF = int (Q xx IF ) dx + C ` i.e., `y xx e ^( 2sqrtx) = int ( e ^(-2sqrtx))/(sqrtx ) xx e ^(2sqrt x ) dx + C ` `" " = int (1)/(sqrtx)dx + C = 2 sqrtx + C`. Hence, ` y e^(2sqrtx) = 2 sqrtx + C ` is the required solution. ` therefore y = (1)/(x) - cot x + (C)/(x sin x ) ` is the required solution.

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