##### 4.6.26 $$a y(x)^2+b x^2 y(x)^3+x^2 y'(x)=0$$

ODE
$a y(x)^2+b x^2 y(x)^3+x^2 y'(x)=0$ ODE Classiﬁcation

[_rational, _Abel]

Book solution method
Abel ODE, Second kind

Mathematica
cpu = 1.08195 (sec), leaf count = 279

$\text {Solve}\left [\frac {\frac {(a y(x)+x) \text {Ai}\left (\frac {x^2+2 a y(x) x+\left (a^2-2 b x^3\right ) y(x)^2}{2 \sqrt [3]{2} a^{2/3} b^{2/3} x^2 y(x)^2}\right )}{2^{2/3} \sqrt [3]{a} \sqrt [3]{b} x y(x)}+\text {Ai}'\left (\frac {x^2+2 a y(x) x+\left (a^2-2 b x^3\right ) y(x)^2}{2 \sqrt [3]{2} a^{2/3} b^{2/3} x^2 y(x)^2}\right )}{\frac {(a y(x)+x) \text {Bi}\left (\frac {x^2+2 a y(x) x+\left (a^2-2 b x^3\right ) y(x)^2}{2 \sqrt [3]{2} a^{2/3} b^{2/3} x^2 y(x)^2}\right )}{2^{2/3} \sqrt [3]{a} \sqrt [3]{b} x y(x)}+\text {Bi}'\left (\frac {x^2+2 a y(x) x+\left (a^2-2 b x^3\right ) y(x)^2}{2 \sqrt [3]{2} a^{2/3} b^{2/3} x^2 y(x)^2}\right )}+c_1=0,y(x)\right ]$

Maple
cpu = 0.195 (sec), leaf count = 178

$\left [y \left (x \right ) = -\frac {2^{\frac {1}{3}} a b x}{2^{\frac {1}{3}} a^{2} b -2 \left (a^{2} b^{2}\right )^{\frac {2}{3}} \RootOf \left (\AiryBi \left (-\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right ) \textit {\_C1} \textit {\_Z} +\textit {\_Z} \AiryAi \left (-\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right )+\AiryBi \left (1, -\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right ) \textit {\_C1} +\AiryAi \left (1, -\frac {b 2^{\frac {2}{3}} x -2 \textit {\_Z}^{2} \left (a^{2} b^{2}\right )^{\frac {1}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}\right )\right ) x}\right ]$ Mathematica raw input

DSolve[a*y[x]^2 + b*x^2*y[x]^3 + x^2*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[C[1] + (AiryAiPrime[(x^2 + 2*a*x*y[x] + (a^2 - 2*b*x^3)*y[x]^2)/(2*2^(1/3)
*a^(2/3)*b^(2/3)*x^2*y[x]^2)] + (AiryAi[(x^2 + 2*a*x*y[x] + (a^2 - 2*b*x^3)*y[x]
^2)/(2*2^(1/3)*a^(2/3)*b^(2/3)*x^2*y[x]^2)]*(x + a*y[x]))/(2^(2/3)*a^(1/3)*b^(1/
3)*x*y[x]))/(AiryBiPrime[(x^2 + 2*a*x*y[x] + (a^2 - 2*b*x^3)*y[x]^2)/(2*2^(1/3)*
a^(2/3)*b^(2/3)*x^2*y[x]^2)] + (AiryBi[(x^2 + 2*a*x*y[x] + (a^2 - 2*b*x^3)*y[x]^
2)/(2*2^(1/3)*a^(2/3)*b^(2/3)*x^2*y[x]^2)]*(x + a*y[x]))/(2^(2/3)*a^(1/3)*b^(1/3
)*x*y[x])) == 0, y[x]]

Maple raw input

dsolve(x^2*diff(y(x),x)+a*y(x)^2+b*x^2*y(x)^3 = 0, y(x))

Maple raw output

[y(x) = -2^(1/3)*a*b*x/(2^(1/3)*a^2*b-2*(a^2*b^2)^(2/3)*RootOf(AiryBi(-1/2*(b*2^
(2/3)*x-2*_Z^2*(a^2*b^2)^(1/3))/(a^2*b^2)^(1/3))*_C1*_Z+_Z*AiryAi(-1/2*(b*2^(2/3
)*x-2*_Z^2*(a^2*b^2)^(1/3))/(a^2*b^2)^(1/3))+AiryBi(1,-1/2*(b*2^(2/3)*x-2*_Z^2*(
a^2*b^2)^(1/3))/(a^2*b^2)^(1/3))*_C1+AiryAi(1,-1/2*(b*2^(2/3)*x-2*_Z^2*(a^2*b^2)
^(1/3))/(a^2*b^2)^(1/3)))*x)]
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