Answer
Nonhomogeneous.
Work Step by Step
General form of second order differential boundary value problem:
$\mathcal{p}$($\mathcal{x}$)$\mathcal{y^{n}}$$\mathcal{+}$$\mathcal{q(x)y^{1}+r(x)y=g(x)}$
$\mathcal{y(x_{0})=y_{0}}$
$\mathcal{y(x_{1})=y_{1}}$
Equation is homogeneous if $\mathcal{g(x)}$ and $\mathcal{y_{0}=y_{1}=0}$
$\mathcal{[(1+x^{2})y^{1}]+4y=0}$
$\mathcal{(1+x^{2})^1y^{1}+(1+x^{2})y^{n}+4y=0}$
$\mathcal{(1+x^{2})y^{n}+2xy^1+4y=0}$
$\mathcal{y(0)=0, y(1)=1}$
Since $\mathcal{y_{1}=1}$, equation is nonhomogeneous