##### 4.6.25 $$x^2 y'(x)=a x^2 y(x)^2-a y(x)^3$$

ODE
$x^2 y'(x)=a x^2 y(x)^2-a y(x)^3$ ODE Classiﬁcation

[_rational, _Abel]

Book solution method
Abel ODE, Second kind

Mathematica
cpu = 0.825741 (sec), leaf count = 239

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

Maple
cpu = 0.11 (sec), leaf count = 117

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

DSolve[x^2*y'[x] == a*x^2*y[x]^2 - a*y[x]^3,y[x],x]

Mathematica raw output

Solve[C[1] + (AiryAiPrime[(x + 2*a*x^2*y[x] + a*(2 + a*x^3)*y[x]^2)/(2*2^(1/3)*a
^(4/3)*x*y[x]^2)] - (AiryAi[(x + 2*a*x^2*y[x] + a*(2 + a*x^3)*y[x]^2)/(2*2^(1/3)
*a^(4/3)*x*y[x]^2)]*(1 + a*x*y[x]))/(2^(2/3)*a^(2/3)*y[x]))/(AiryBiPrime[(x + 2*
a*x^2*y[x] + a*(2 + a*x^3)*y[x]^2)/(2*2^(1/3)*a^(4/3)*x*y[x]^2)] - (AiryBi[(x +
2*a*x^2*y[x] + a*(2 + a*x^3)*y[x]^2)/(2*2^(1/3)*a^(4/3)*x*y[x]^2)]*(1 + a*x*y[x]
))/(2^(2/3)*a^(2/3)*y[x])) == 0, y[x]]

Maple raw input

dsolve(x^2*diff(y(x),x) = a*x^2*y(x)^2-a*y(x)^3, y(x))

Maple raw output

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