\[ 2 a x^3 y(x)^3+y'(x)+2 x y(x)=0 \] ✓ Mathematica : cpu = 0.0725068 (sec), leaf count = 72
\[\left \{\left \{y(x)\to -\frac {\sqrt {2}}{\sqrt {-2 a x^2-a+2 c_1 e^{2 x^2}}}\right \},\left \{y(x)\to \frac {\sqrt {2}}{\sqrt {-2 a x^2-a+2 c_1 e^{2 x^2}}}\right \}\right \}\] ✓ Maple : cpu = 0.017 (sec), leaf count = 53
\[ \left \{ y \left ( x \right ) =-2\,{\frac {1}{\sqrt {-4\,a{x}^{2}+4\,{{\rm e}^{2\,{x}^{2}}}{\it \_C1}-2\,a}}},y \left ( x \right ) =2\,{\frac {1}{\sqrt {-4\,a{x}^{2}+4\,{{\rm e}^{2\,{x}^{2}}}{\it \_C1}-2\,a}}} \right \} \]
\begin {equation} y^{\prime }=-2xy-2ax^{3}y^{3}\tag {1} \end {equation}
This is of the form \(y^{\prime }=f_{0}+f_{1}y+f_{2}y^{2}+f_{3}y^{3}\) where \(f_{0}=0,f_{2}=0\). Hence this is Bernoulli first order non-linear ODE. We start by diving by \(y^{3}\)\[ \frac {y^{\prime }}{y^{3}}=-2x\frac {1}{y^{2}}-2ax^{3}\] Let \(u=\frac {1}{y^{2}}\), hence \(u^{\prime }=-2\frac {y^{\prime }}{y^{3}}\) and the above becomes\begin {align*} -\frac {1}{2}u^{\prime } & =-2xu-2ax^{3}\\ u^{\prime }-4xu & =4ax^{3} \end {align*}
Integrating factor is \(e^{-4\int xdx}=e^{-2x^{2}}\) hence\[ \frac {d}{dx}\left ( e^{-2x^{2}}u\right ) =4ax^{3}e^{-2x^{2}}\] Integrating\begin {align*} e^{-2x^{2}}u & =4a\int x^{3}e^{-2x^{2}}dx+C\\ & =4a\left ( \frac {-1}{8}\left ( 2x^{2}+1\right ) e^{-2x^{2}}\right ) +C \end {align*}
Therefore\[ u=-\frac {1}{2}a\left ( 2x^{2}+1\right ) +Ce^{2x^{2}}\] Hence\[ y^{2}=\frac {1}{u}=\frac {1}{-\frac {1}{2}a\left ( 2x^{2}+1\right ) +Ce^{2x^{2}}}\] Or\[ y=\pm \frac {\sqrt {2}}{\sqrt {-a\left ( 2x^{2}+1\right ) +Ce^{2x^{2}}}}\] Verification
ode:=2*a*x^3*y(x)^3+diff(y(x),x)+2*x*y(x)=0; my_sol:=sqrt(2)/sqrt(-a*(2*x^2+1)+_C1*exp(2*x^2)); odetest(y(x)=my_sol,ode); 0 my_sol:=-sqrt(2)/sqrt(-a*(2*x^2+1)+_C1*exp(2*x^2)); odetest(y(x)=my_sol,ode); 0