Solved using ﬁrst_order_ode_riccati_by_guessing_particular

Entering ﬁrst order ode riccati guess solver

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

This is a Riccati ODE. Comparing the above ODE to solve with the Riccati standard form

\[ n' = f_0(x)+ f_1(x)n+f_2(x)n^{2} \]

Shows that

\begin{align*} f_0(x) & =x\\ f_1(x) & =0\\ f_2(x) &=x \end{align*}

Using trial and error, the following particular solution was found

\begin{align*} n_p &= i \end{align*}

Since a particular solution is known, then the general solution is given by

\begin{align*} n &= n_p + \frac { \phi (x) }{ c_1 - \int { \phi (x) f_2 \,dx} } \end{align*}

Where

\begin{align*} \phi (x) &= e^{ \int 2 f_2 n_p + f_1 \,dx } \end{align*}

Evaluating the above gives the general solution as

\begin{align*} n = \frac {{\mathrm e}^{i x^{2}}+2 i c_1}{i {\mathrm e}^{i x^{2}}+2 c_1} \end{align*}


Direction ﬁeld with Isoclines	Direction ﬁeld

Summary of solutions found

\[ n = \frac {{\mathrm e}^{i x^{2}}+2 i c_1}{i {\mathrm e}^{i x^{2}}+2 c_1} \]