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 = 0.871592 (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 = 0.095 (sec), leaf count = 308

\[ \left \{ {\it \_C1}+{1 \left ( a{{\rm Ai}\left ( \left ( {\frac {\sqrt [3]{2}}{2\,bx}\sqrt [3]{{a}^{2}{b}^{2}}}+{\frac {ab\sqrt [3]{2}}{2\,y \left ( x \right ) } \left ( {a}^{2}{b}^{2} \right ) ^{-{\frac {2}{3}}}} \right ) ^{2}-{\frac {b{2}^{{\frac {2}{3}}}x}{2}{\frac {1}{\sqrt [3]{{a}^{2}{b}^{2}}}}}\right )}b\sqrt [3]{2} \left ( x+ay \left ( x \right ) \right ) +2\, \left ( {a}^{2}{b}^{2} \right ) ^{2/3}{{\rm Ai}^{(1)}\left ( \left ( 1/2\,{\frac {\sqrt [3]{2}\sqrt [3]{{a}^{2}{b}^{2}}}{bx}}+1/2\,{\frac {ab\sqrt [3]{2}}{ \left ( {a}^{2}{b}^{2} \right ) ^{2/3}y \left ( x \right ) }} \right ) ^{2}-1/2\,{\frac {b{2}^{2/3}x}{\sqrt [3]{{a}^{2}{b}^{2}}}}\right )}xy \left ( x \right ) \right ) \left ( a{{\rm Bi}\left ( \left ( {\frac {\sqrt [3]{2}}{2\,bx}\sqrt [3]{{a}^{2}{b}^{2}}}+{\frac {ab\sqrt [3]{2}}{2\,y \left ( x \right ) } \left ( {a}^{2}{b}^{2} \right ) ^{-{\frac {2}{3}}}} \right ) ^{2}-{\frac {b{2}^{{\frac {2}{3}}}x}{2}{\frac {1}{\sqrt [3]{{a}^{2}{b}^{2}}}}}\right )}b\sqrt [3]{2} \left ( x+ay \left ( x \right ) \right ) +2\, \left ( {a}^{2}{b}^{2} \right ) ^{2/3}{{\rm Bi}^{(1)}\left ( \left ( 1/2\,{\frac {\sqrt [3]{2}\sqrt [3]{{a}^{2}{b}^{2}}}{bx}}+1/2\,{\frac {ab\sqrt [3]{2}}{ \left ( {a}^{2}{b}^{2} \right ) ^{2/3}y \left ( x \right ) }} \right ) ^{2}-1/2\,{\frac {b{2}^{2/3}x}{\sqrt [3]{{a}^{2}{b}^{2}}}}\right )}xy \left ( x \right ) \right ) ^{-1}}=0 \right \} \] 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),'implicit')

Maple raw output

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