Let’s consider the second order linear homogeneous differential equation

\[\displaystyle y\,^{\prime \prime }+\dfrac {3}{x}y\,^{\prime }+\dfrac {1}{x^2}y=0.\]
(a)
Suppose \(y_1\) and \(y_2\) are a fundamental set of solutions to this differential equation on \((0,\infty )\) and that their Wronskian \(W(x)\) satisfies \(W(1)=1\). Let’s determine \(W(x)\).
(b)
Let’s show that \(y_1=x^{-1}\ln {(x)}\) is a solution to this differential equation.
(c)
It turns out that \(y_2=x^{-1}\ln {(x^2)}\) is also a solution. Does \(\{y_1,y_2\}\) form a fundamental set of solutions to \(\displaystyle y\,^{\prime \prime }+\dfrac {3}{x}y\,^{\prime }+\dfrac {1}{x^2}y=0\) on \((0,\infty )\)?
(d)
It turns out that with \(y_2=x^{-1}\) we get a fundamental set of solutions. Let’s solve the IVP \(\displaystyle y^{\,\prime \prime }+\dfrac {3}{x}y^{\,\prime }+\dfrac {1}{x^2}y=0\), \(y(1)=1\), \(y^{\,\prime }(1)=0\).