2.2.3.5 ✓ Maple. Time used: 0.001 (sec). Leaf size: 17

ode:=y(x)-x*diff(y(x),x) = b*(1+x^2*diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y = \frac {c_1 x +b}{b x +1} \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right )-x \left (\frac {d}{d x}y \left (x \right )\right )=b \left (1+x^{2} \left (\frac {d}{d x}y \left (x \right )\right )\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {-y \left (x \right )+b}{-b \,x^{2}-x} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d x}y \left (x \right )}{-y \left (x \right )+b}=\frac {1}{-b \,x^{2}-x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\frac {d}{d x}y \left (x \right )}{-y \left (x \right )+b}d x =\int \frac {1}{-b \,x^{2}-x}d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (-y \left (x \right )+b \right )=-\ln \left (x \right )+\ln \left (b x +1\right )+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )=\frac {{\mathrm e}^{\mathit {C1}} b^{2} x +{\mathrm e}^{\mathit {C1}} b -x}{{\mathrm e}^{\mathit {C1}} \left (b x +1\right )} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y \left (x \right )=\frac {-{\mathrm e}^{-\mathit {C1}} x +b^{2} x +b}{b x +1} \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & y \left (x \right )=\frac {b^{2} x +\mathit {C1} x +b}{b x +1} \end {array} \]