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 = 0.617778 (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 = 0.081 (sec), leaf count = 182

\[ \left \{ {\it \_C1}+{1 \left (\left (-axy \relax (x ) -1 \right ) {{\rm Ai}\left (\left ({\frac {x}{2}\sqrt [3]{-2\,a}}-{\frac {1}{y \relax (x ) } \left (-2\,a \right ) ^{-{\frac {2}{3}}}} \right ) ^{2}-{\frac {1}{x}{\frac {1}{\sqrt [3]{-2\,a}}}}\right )}+{{\rm Ai}^{(1)}\left (\left ({\frac {x}{2}\sqrt [3]{-2\,a}}-{\frac {1}{y \relax (x ) } \left (-2\,a \right ) ^{-{\frac {2}{3}}}} \right ) ^{2}-{\frac {1}{x}{\frac {1}{\sqrt [3]{-2\,a}}}}\right )} \left (-2\,a \right ) ^{{\frac {2}{3}}}y \relax (x ) \right ) \left (\left (-axy \relax (x ) -1 \right ) {{\rm Bi}\left (\left ({\frac {x}{2}\sqrt [3]{-2\,a}}-{\frac {1}{y \relax (x ) } \left (-2\,a \right ) ^{-{\frac {2}{3}}}} \right ) ^{2}-{\frac {1}{x}{\frac {1}{\sqrt [3]{-2\,a}}}}\right )}+{{\rm Bi}^{(1)}\left (\left ({\frac {x}{2}\sqrt [3]{-2\,a}}-{\frac {1}{y \relax (x ) } \left (-2\,a \right ) ^{-{\frac {2}{3}}}} \right ) ^{2}-{\frac {1}{x}{\frac {1}{\sqrt [3]{-2\,a}}}}\right )} \left (-2\,a \right ) ^{{\frac {2}{3}}}y \relax (x ) \right ) ^{-1}}=0 \right \} \] 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),'implicit')

Maple raw output

_C1+((-a*x*y(x)-1)*AiryAi((1/2*(-2*a)^(1/3)*x-1/(-2*a)^(2/3)/y(x))^2-1/(-2*a)^(1
/3)/x)+AiryAi(1,(1/2*(-2*a)^(1/3)*x-1/(-2*a)^(2/3)/y(x))^2-1/(-2*a)^(1/3)/x)*(-2
*a)^(2/3)*y(x))/((-a*x*y(x)-1)*AiryBi((1/2*(-2*a)^(1/3)*x-1/(-2*a)^(2/3)/y(x))^2
-1/(-2*a)^(1/3)/x)+AiryBi(1,(1/2*(-2*a)^(1/3)*x-1/(-2*a)^(2/3)/y(x))^2-1/(-2*a)^
(1/3)/x)*(-2*a)^(2/3)*y(x)) = 0