2.15.1 Example lines are not parallel

\[ y^{\prime }=\frac {-6x+y-3}{2x-y-1}\]

Comparing to \(y^{\prime }=\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}\) shows that \(a_{1}=-6,b_{1}=1,a_{2}=2,b_{2}=-1\). Hence \(\frac {a_{1}}{b_{1}}=-6,\frac {a_{2}}{b_{2}}=-2\). This shows the lines are not parallel. Let

\begin{align*} X & =x-x_{0}\\ Y & =y-y_{0}\end{align*}

The constant \(x_{0},y_{0}\) are found by solving

\begin{align*} a_{1}x_{0}+b_{1}y_{0}+c_{1} & =0\\ a_{2}x_{0}+b_{2}y_{0}+c_{2} & =0 \end{align*}

Or

\begin{align*} -6x_{0}+y_{0}-3 & =0\\ 2x_{0}-y_{0}-1 & =0 \end{align*}

Solving for \(x_{0},y_{0}\) gives

\begin{align*} x_{0} & =-1\\ y_{0} & =-3 \end{align*}

Hence

\begin{align*} X & =x+1\\ Y & =y+3 \end{align*}

Using this transformation in \(y^{\prime }=\frac {-6x+y-3}{2x-y-1}\) results in the ode

\[ \frac {dY}{dX}=\frac {6X-Y}{-2X+Y}\]

This is a homogeneous ode

\[ \frac {dY}{dX}=\frac {6-\frac {Y}{X}}{-2+\frac {Y}{Y}}\]

Let \(u=\frac {Y}{X}\). Now it is solved as was shown in the above sections. At the end, \(Y\) is replaced by \(y-y_{0}\) to obtain the solution in \(y\left ( x\right ) \).