Let’s consider the second-order linear non-homogeneous ordinary differential equation \(xy^{\,\prime \prime }-y^{\,\prime }-4x^3y=8x^5\) on the interval \((0,\infty )\).

(a)
Let’s show that \(y_1=e^{x^2}\) and \(y_2=e^{-x^2}\) are linearly independent solutions to the complementary equation.
(b)
Let’s determine a particular solution to this differential equation using Variation of Parameters.
(c)
Let’s determine the general solution to this differential equation.
(d)
Let’s solve the IVP \(xy^{\,\prime \prime }-y^{\,\prime }-4x^3y=8x^5\), \(y(1)=0\), \(y^{\,\prime }(1)=0\).