#### 2.18   ODE No. 18

$y'(x)-y(x)^2-x y(x)-x+1=0$ Mathematica : cpu = 0.074026 (sec), leaf count = 50

$\left \{\left \{y(x)\to -1+\frac {e^{\frac {x^2}{2}-2 x}}{-\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {x-2}{\sqrt {2}}\right )}{e^2}+c_1}\right \}\right \}$ Maple : cpu = 0.051 (sec), leaf count = 39

$\left \{ y \left ( x \right ) =-1+{\frac {1}{{\it \_C1}+{\frac {i}{2}}\sqrt {\pi }{{\rm e}^{-2}}\sqrt {2}{\it Erf} \left ( {\frac {i}{2}}\sqrt {2} \left ( x-2 \right ) \right ) }{{\rm e}^{{\frac {x \left ( x-4 \right ) }{2}}}}} \right \}$

Hand solution

\begin {align} y^{\prime }-y^{2}-xy-x+1 & =0\nonumber \\ y^{\prime } & =x-1+xy+y^{2}\tag {1} \end {align}

This is Riccati ﬁrst order non-linear ODE of the form. The general form is$y^{\prime }=P\left ( x\right ) +Q\left ( x\right ) y+R\left ( x\right ) y^{2}$ Where $$P\left ( x\right ) =x-1,Q\left ( x\right ) =x,R\left ( x\right ) =1$$. We see that $$y_{p}=-1$$ is a particular solution, therefore we use the substitution $$y=y_{p}+\frac {1}{u}$$, hence $$y^{\prime }=-\frac {u^{\prime }}{u^{2}}$$ and equating this to (1) we obtain\begin {align*} -\frac {u^{\prime }}{u^{2}} & =x-1+xy+y^{2}\\ & =x-1+x\left ( -1+\frac {1}{u}\right ) +\left ( -1+\frac {1}{u}\right ) ^{2}\\ & =x-1-x+\frac {x}{u}+\left ( 1+\frac {1}{u^{2}}-\frac {2}{u}\right ) \\ & =\frac {x}{u}+\frac {1}{u^{2}}-\frac {2}{u} \end {align*}

Hence\begin {align*} u^{\prime } & =-u^{2}\left ( \frac {x}{u}+\frac {1}{u^{2}}-\frac {2}{u}\right ) \\ & =-xu-1+2u\\ u^{\prime }+xu-2u & =-1\\ u^{\prime }+u\left ( x-2\right ) & =-1 \end {align*}

Integration factor is $$e^{\int \left ( x-2\right ) dx}=e^{\frac {x^{2}}{2}-2x}=e^{\frac {1}{2}x\left ( x-4\right ) }$$, therefore$d\left ( e^{\frac {1}{2}x\left ( x-4\right ) }u\right ) =-e^{\frac {1}{2}x\left ( x-4\right ) }$ Integrating both sides$e^{\frac {1}{2}x\left ( x-4\right ) }u=-\int e^{\frac {1}{2}x\left ( x-4\right ) }+C$ But $\int e^{\frac {1}{2}x\left ( x-4\right ) }=\frac {1}{e^{2}}\sqrt {\frac {\pi }{2}}\operatorname {erfi}\left ( \frac {x-2}{\sqrt {2}}\right )$ Hence$u\left ( x\right ) =e^{\frac {-1}{2}x\left ( x-4\right ) }\left ( \frac {-1}{e^{2}}\sqrt {\frac {\pi }{2}}\operatorname {erfi}\left ( \frac {x-2}{\sqrt {2}}\right ) +C\right )$ Since $$y=y_{p}+\frac {1}{u}$$ then$y=-1+\frac {1}{e^{\frac {-1}{2}x\left ( x-4\right ) }\left ( \frac {-1}{e^{2}}\sqrt {\frac {\pi }{2}}\operatorname {erfi}\left ( \frac {x-2}{\sqrt {2}}\right ) +C\right ) }$ Or$y=\frac {e^{\frac {1}{2}x\left ( x-4\right ) }}{C-\frac {1}{e^{2}}\sqrt {\frac {\pi }{2}}\operatorname {erfi}\left ( \frac {x-2}{\sqrt {2}}\right ) }-1$