First of all, we say that y and \(\displaystyle{z}={y}{v}\) are linearly independent if:

\(\displaystyle{A}{y}+{B}{z}={0}\forall\Rightarrow{A}={B}={0}\)

Notice that:

Notice that:

\(\displaystyle{A}{y}+{B}{y}={y}{\left({A}+{B}{v}\right)}\)

If v is constant for all t, then

\(\displaystyle{A}=-{B}{v}\Rightarrow{A}{y}+{B}{y}{v}={0}\)

Therefore, we need that v is not constant over time in order to have that y and z are linearly independent.

Since \(\displaystyle{y}{\left({t}\right)}\ {i}{s}\ {a}\ {s}{o}{l}{u}{t}{i}{o}{n}\ {o}{f}\ \ddot{{{x}}}+{a}{\left({t}\right)}\dot{{{x}}}+{b}{\left({t}\right)}{x}={0}\), then:

\(\displaystyle\ddot{{{y}}}+{a}{\left({t}\right)}\dot{{{y}}}+{b}{\left({t}\right)}{y}={0}\)

Let's consider \(\displaystyle{z}{\left({t}\right)}={y}{\left({t}\right)}{v}{\left({t}\right)}.{I}{f}\ {z}{\left({t}\right)}\ {i}{s}\ {a}\ {s}{o}{l}{u}{t}{i}{o}{n}\ {o}{f}\ \ddot{{{x}}}+{a}{\left({t}\right)}\dot{{{x}}}+{b}{\left({t}\right)}{x}={0}\) then:

\(\displaystyle\ddot{{{z}}}+{a}{\left({t}\right)}\dot{{{z}}}+{b}{\left({t}\right)}{z}={0}\Rightarrow\)

\(\displaystyle{\left(\ddot{{{y}}}{v}\right)}+{a}{\left({t}\right)}{\left(\dot{{{y}}}{v}\right)}+{b}{\left({t}\right)}{y}{v}={0}\Rightarrow\)

\(\displaystyle\ddot{{{y}}}{v}+{2}\dot{{{y}}}\dot{{{v}}}+{y}\ddot{{{v}}}+{a}{\left({t}\right)}{\left(\dot{{{y}}}{v}+{y}\dot{{{v}}}\right)}+{b}{\left({t}\right)}{y}{v}={0}\Rightarrow\)

\(\displaystyle{v}{\left(\ddot{{{y}}}+{a}{\left({t}\right)}\dot{{{y}}}+{b}{\left({t}\right)}{y}\right)}+{2}\dot{{{y}}}\dot{{{v}}}+{a}{\left({t}\right)}{y}\dot{{{v}}}={0}\)

\(\displaystyle{v}\cdot{0}+{2}\dot{{{y}}}\dot{{{v}}}+{y}\ddot{{{v}}}+{a}{\left({t}\right)}{y}\dot{{{v}}}={0}\)

\(\displaystyle{y}\ddot{{{v}}}+{\left({2}\dot{{{y}}}+{a}{\left({t}\right)}{y}\right)}\dot{{{v}}}={0}\)

Now let's introduce \(\displaystyle{w}=\dot{{{v}}}\). The previous equation can be rewritten as:

\(\displaystyle{\left\lbrace\begin{array}{c} \dot{{{v}}}={w}\\{y}\dot{{{w}}}=-{\left({2}\dot{{{y}}}+{a}{\left({t}\right)}{y}\right)}{w}\end{array}\right.}\)

This means that the non constant function v satisfies the previous system of ODEs.