2.2.3.1 Solved using first_order_ode_linear

Entering first order ode linear solver

\[\begin {aligned} y-y^{\prime } x&=b \left (1+y^{\prime } x^{2}\right ) \end {aligned}\]

In canonical form a linear first order is

\begin{align*} y^{\prime } + q(x)y &= p(x) \end{align*}

Comparing the above to the given ode shows that

\begin{align*} q(x) &=-\frac {1}{x \left (b x +1\right )}\\ p(x) &=-\frac {b}{x \left (b x +1\right )} \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int -\frac {1}{x \left (b x +1\right )}d x}\\ &= \frac {b x +1}{x} \end{align*}

The ode becomes

\begin{gather*} \begin {aligned} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \mu p \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (-\frac {b}{x \left (b x +1\right )}\right )\\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\frac {y \left (b x +1\right )}{x}\right ) &= \left (\frac {b x +1}{x}\right ) \left (-\frac {b}{x \left (b x +1\right )}\right )\\ \mathrm {d} \left (\frac {y \left (b x +1\right )}{x}\right ) &= \left (-\frac {b}{x^{2}}\right )\, \mathrm {d} x \end {aligned} \end{gather*}

Integrating gives

\begin{align*} \frac {y \left (b x +1\right )}{x}&= \int {-\frac {b}{x^{2}} \,dx} \\ &=\frac {b}{x} + c_1 \end{align*}

Dividing throughout by the integrating factor \(\frac {b x +1}{x}\) gives the final solution

\[ y = \frac {c_1 x +b}{b x +1} \]

Summary of solutions found

\[ y = \frac {c_1 x +b}{b x +1} \]