#### 2.36   ODE No. 36

$a x y(x)^2+y'(x)+y(x)^3=0$ Mathematica : cpu = 0.258517 (sec), leaf count = 195

$\text {Solve}\left [\frac {\text {Ai}'\left (\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} a^{4/3} x^2\right )-\left (-\frac {1}{2}\right )^{2/3} a^{2/3} x \text {Ai}\left (\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} a^{4/3} x^2\right )}{\text {Bi}'\left (\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} a^{4/3} x^2\right )-\left (-\frac {1}{2}\right )^{2/3} a^{2/3} x \text {Bi}\left (\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{a}}{y(x)}-\frac {1}{2} \sqrt [3]{-\frac {1}{2}} a^{4/3} x^2\right )}+c_1=0,y(x)\right ]$ Maple : cpu = 0.073 (sec), leaf count = 62

$\left \{ y \left ( x \right ) =2\,{\frac {a}{{a}^{2}{x}^{2}+2\,{\it RootOf} \left ( {{\rm Bi}\left ({\it \_Z}\right )}\sqrt [3]{-2\,{a}^{2}}{\it \_C1}\,x+\sqrt [3]{-2\,{a}^{2}}x{{\rm Ai}\left ({\it \_Z}\right )}+2\,{{\rm Bi}^{(1)}\left ({\it \_Z}\right )}{\it \_C1}+2\,{{\rm Ai}^{(1)}\left ({\it \_Z}\right )} \right ) \sqrt [3]{-2\,{a}^{2}}}} \right \}$

Hand solution

\begin {equation} y^{\prime }\left ( x\right ) =-axy^{2}-y^{3}\tag {1} \end {equation}

This is Abel ﬁrst order non-linear. The general form is of Abel ﬁrst kind is$y^{\prime }\left ( x\right ) =f_{0}\left ( x\right ) +f_{1}\left ( x\right ) y\left ( x\right ) +f_{2}\left ( x\right ) y^{2}\left ( x\right ) +f_{3}\left ( x\right ) y^{3}\left ( x\right )$ In this case, $$f_{0}\left ( x\right ) =0,f_{1}\left ( x\right ) =0,f_{2}\left ( x\right ) =-ax,f_{3}\left ( x\right ) =-1$$. Note $$\left ( \frac {f_{3}}{f_{2}}\right ) ^{\prime }=\left ( \frac {1}{ax}\right ) ^{\prime }=-\frac {1}{a}$$. While Abel second kind has the form$\left ( y+g\left ( x\right ) \right ) y^{\prime }\left ( x\right ) =f_{0}\left ( x\right ) +f_{1}\left ( x\right ) y\left ( x\right ) +f_{2}\left ( x\right ) y^{2}\left ( x\right )$ For $$g\left ( x\right ) \neq 0$$.

Looking at (1) again, using the transformation suggested in Kamke $$u=\frac {1}{y}-\frac {1}{2}ax^{2}$$ or $$y=\frac {1}{u+\frac {1}{2}ax^{2}}$$Then$y^{\prime }=\frac {-u^{\prime }-ax}{\left ( u+\frac {1}{2}ax^{2}\right ) ^{2}}$

Equating the above to the RHS of (1) gives\begin {align*} \frac {-u^{\prime }-ax}{\left ( u+\frac {1}{2}ax^{2}\right ) ^{2}} & =-ax\left ( \frac {1}{u+\frac {1}{2}ax^{2}}\right ) ^{2}-\left ( \frac {1}{u+\frac {1}{2}ax^{2}}\right ) ^{3}\\ -u^{\prime }-ax & =-ax-\frac {1}{u+\frac {1}{2}ax^{2}}\\ \frac {du}{dx} & =\frac {1}{u+\frac {1}{2}ax^{2}} \end {align*}

Writing as\begin {equation} \frac {dx}{du}=u+\frac {1}{2}ax^{2}\tag {2} \end {equation} This can now be viewed as reverse Riccati in $$x$$. Using the standard transformation\begin {equation} x=-\frac {z^{\prime }}{z\left ( \frac {1}{2}a\right ) }=-\frac {2z^{\prime }}{az}\tag {3} \end {equation} Hence$\frac {dx}{du}=-\frac {2}{a}\left ( \frac {z^{\prime \prime }}{z}-\frac {\left ( z^{\prime }\right ) ^{2}}{z^{2}}\right )$ Equating this to RHS of (2) gives a second order Airy ODE where the dependent variable is $$z$$ and the independent variable is $$u$$\begin {align*} -\frac {2}{a}\left ( \frac {z^{\prime \prime }}{z}-\frac {\left ( z^{\prime }\right ) ^{2}}{z^{2}}\right ) & =u+\frac {1}{2}a\left ( -\frac {2z^{\prime }}{az}\right ) ^{2}\\ -\frac {2}{a}\frac {z^{\prime \prime }}{z}+\frac {2}{a}\frac {\left ( z^{\prime }\right ) ^{2}}{z^{2}} & =u+\frac {1}{2}a\frac {4\left ( z^{\prime }\right ) ^{2}}{a^{2}z^{2}}\\ -\frac {2}{a}\frac {z^{\prime \prime }}{z}+\frac {2}{a}\frac {\left ( z^{\prime }\right ) ^{2}}{z^{2}} & =u+\frac {2}{a}\frac {\left ( z^{\prime }\right ) ^{2}}{z^{2}}\\ -\frac {2}{a}\frac {z^{\prime \prime }}{z} & =u\\ z^{\prime \prime }\left ( u\right ) +\frac {a}{2}uz\left ( u\right ) & =0 \end {align*}

This is Airy ODE whose solution is found using power series method. The solution is\begin {equation} z\left ( u\right ) =C_{1}\operatorname {AiryAI}\left ( -\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) +C_{2}\operatorname {AiryBI}\left ( -\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) \tag {4} \end {equation} We now go back to (3) and ﬁnd $$x$$$x=-\frac {2z^{\prime }}{az}$ Since \begin {align*} \frac {d}{du}\operatorname {AiryAI}\left ( -\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) & =-\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}\operatorname {AiryAI}\left ( 1,-\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) \\ \frac {d}{du}\operatorname {AiryBI}\left ( -\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) & =-\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}\operatorname {AiryBI}\left ( 1,-\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) \end {align*}

Then $x=-\frac {2}{a}\frac {-C_{1}\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}\operatorname {AiryAI}\left ( 1,-\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) -C_{2}\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}\operatorname {AiryBI}\left ( 1,-\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) }{C_{1}\operatorname {AiryAI}\left ( -\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) +C_{2}\operatorname {AiryBI}\left ( -\frac {1}{2}2^{\frac {2}{3}}a^{\frac {1}{3}}u\right ) }$ Therefore $$\frac {dx}{du}$$ is now found from above. Once we ﬁnd $$\frac {dx}{du}$$ then $$\frac {du}{dx}$$ is also found. Using $$\frac {du}{dx}=\frac {1}{u+\frac {1}{2}ax^{2}}$$ now $$u\left ( x\right )$$ is found. Once $$u\left ( x\right )$$ is found then $$y\left ( x\right )$$ is found from the original transformation $$y=\frac {1}{u+\frac {1}{2}ax^{2}}$$. This is all now just algebra.