Let y=y(x) be the solution of the differential equation,
`(x^(2)+1)^(2)dy/dx+2x(x^(2)+1)y=1` such that y(0) =0. If
`sqrta y(1)=pi/32`,then tha value of 'a' is

Explanation

`(x^2+1)^2(dy)/(dx)+2x(x^2+1)y=1` Liner differential equation `implies(dy)/(dx)+((2x)/(x^2+1))y=1/((x^2+1)^2)`
Integrating factor `=e^(int(2x)/(1+x^2)dx)`
Let `x^2+1=trArr2xdx=dt`
`=e^(int(dt)/(t))=e^("ln t)=t=x^2+1`
`impliesy(x^2+1)=int(dx)/(x^2+1)=tan^(-1)x+Cimpliesy(x^2+1)=tan^(-1)x+C`
`:'x=0rArry=0`
`0=tan^(-1)0+CrarrC=0`
`y(x^2+1)=tan^(-1)xrArry=(tan^(-1)x)/(x^2+1)`
`y(1)=pi/8,sqrt(a)y(1)=pi/32`
`impliessqrt(a)xxpi/8=pi/32rArrsqrt(a)=1/4rArra=1/16`