2.6.10.1 Solved using first_order_ode_linear

Entering first order ode linear solver

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

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) &=-\left (\sqrt {x}\, \sin \left (x^{2}-1\right )-2\right ) \sqrt {x}\\ p(x) &=0 \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int -\left (\sqrt {x}\, \sin \left (x^{2}-1\right )-2\right ) \sqrt {x}d x}\\ &= {\mathrm e}^{\frac {\cos \left (x^{2}-1\right )}{2}+\frac {4 x^{{3}/{2}}}{3}} \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \mu y &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (y \,{\mathrm e}^{\frac {\cos \left (x^{2}-1\right )}{2}+\frac {4 x^{{3}/{2}}}{3}}\right ) &= 0 \end{align*}

Integrating gives

\begin{align*} y \,{\mathrm e}^{\frac {\cos \left (x^{2}-1\right )}{2}+\frac {4 x^{{3}/{2}}}{3}}&= \int {0 \,dx} + c_1 \\ &=c_1 \end{align*}

Dividing throughout by the integrating factor \({\mathrm e}^{\frac {\cos \left (x^{2}-1\right )}{2}+\frac {4 x^{{3}/{2}}}{3}}\) gives the final solution

\[ y = {\mathrm e}^{-\frac {\cos \left (x^{2}-1\right )}{2}-\frac {4 x^{{3}/{2}}}{3}} c_1 \]

Direction field with Isoclines	Direction field

Summary of solutions found

\[ y = {\mathrm e}^{-\frac {\cos \left (x^{2}-1\right )}{2}-\frac {4 x^{{3}/{2}}}{3}} c_1 \]