#### 2.102   ODE No. 102

$-a x^3+x y'(x)+x y(x)^2-y(x)=0$ Mathematica : cpu = 0.0847245 (sec), leaf count = 36

$\left \{\left \{y(x)\to \sqrt {a} x \tanh \left (\frac {1}{2} \left (2 \sqrt {a} c_1+\sqrt {a} x^2\right )\right )\right \}\right \}$ Maple : cpu = 0.031 (sec), leaf count = 22

$\left \{ y \left ( x \right ) =\tanh \left ( {\frac {{x}^{2}+2\,{\it \_C1}}{2}\sqrt {a}} \right ) x\sqrt {a} \right \}$

Hand solution

\begin {equation} xy^{\prime }+xy^{2}-y-ax^{3}=0\nonumber \end {equation} This is Riccati non-linear ﬁrst order. But using the transformation $$y=xv$$ it is transformed to easily solved ODE$y^{\prime }=v+xv^{\prime }$

Therefore the ODE becomes

\begin {align*} x\left ( v+xv^{\prime }\right ) +x\left ( xv\right ) ^{2}-xv-ax^{3} & =0\\ xv+x^{2}v^{\prime }+x^{3}v^{2}-xv-ax^{3} & =0\\ x^{2}v^{\prime }+x^{3}v^{2}-ax^{3} & =0\\ v^{\prime }+xv^{2}-ax & =0\\ \frac {dv}{dx} & =x\left ( a-v^{2}\right ) \\ \frac {dv}{a-v^{2}} & =xdx \end {align*}

Integrating

\begin {align*} \frac {1}{\sqrt {a}}\tanh ^{-1}\left ( \frac {v}{\sqrt {a}}\right ) & =\frac {x^{2}}{2}+C\\ \tanh ^{-1}\left ( \frac {v}{\sqrt {a}}\right ) & =\sqrt {a}\left ( \frac {x^{2}}{2}+C\right ) \\ \frac {v}{\sqrt {a}} & =\tanh \left ( \sqrt {a}\left ( \frac {x^{2}}{2}+C\right ) \right ) \\ v & =\sqrt {a}\tanh \left ( \sqrt {a}\left ( \frac {x^{2}}{2}+C\right ) \right ) \end {align*}

Therefore

\begin {align*} y & =xv\\ & =x\sqrt {a}\tanh \left ( \sqrt {a}\left ( \frac {x^{2}}{2}+C\right ) \right ) \end {align*}

Veriﬁcation

restart;
ode:=x*diff(y(x),x)+x*y(x)^2-y(x)-a*x^3=0;
my_solution:=x*sqrt(a)*tanh(sqrt(a)*(x^2/2+_C1));
odetest(y(x)=my_solution,ode);
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