Verify that `y=c e^(t a n-1_x)` is a solution of differential equation `(1+x^2)(d^2y)/(dx^2)+x(dy)/(dx)=0.`

Explanation

`y=c*e^(tan^(-1)x)` …………`(1)`
Differentiate with respect to `x`
`(dy)/(dx)=c*e^(tan^(-1)x)*(1)/(1+x^(2))`
`implies (1+x^(2))(dy)/(dx)=c*e^(tan^(-1)x)`
`implies (1+x^(2))(dy)/(dx)=y` [From eq. `(1)`]
Again differentiate with respect to `x`
`(1+x^(2))(d^(2)y)/(dx^(2))+(dy)/(Dx)*2x=(dy)/(dx)`
`implies (1+x^(2))(d^(2)y)/(dx^(2))+(2x-1)(dy)/(dx)=0`
`:. y=c*e^(tan^(-1)x)` is a solution of the given differential eqaution.