4.6.26 $ay(x{)}^2+b{x}^{2}y(x{)}^3+x^2{y}^{\prime }(x)=0$

ODE
$$ay(x{)}^2+b{x}^{2}y(x{)}^3+x^2{y}^{\prime }(x)=0$$ ODE Classification

[_rational, _Abel]

Book solution method
Abel ODE, Second kind

Mathematica ✓
cpu = 0.871592 (sec), leaf count = 279

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

Maple ✓
cpu = 0.095 (sec), leaf count = 308

$$\{\mathit{\_}\mathit{C}\mathit{1}+1(a\mathrm{A}\mathrm{i}({(\frac{\sqrt[3]{2}}{2bx}\sqrt[3]{a^2{b}^2}+\frac{ab\sqrt[3]{2}}{2y\left(x\right)}{\left(a^2{b}^2\right)}^{-\frac{2}{3}})}^2-\frac{b{2}^{\frac{2}{3}}x}{2}\frac{1}{\sqrt[3]{a^2{b}^2}})b\sqrt[3]{2}(x+ay\left(x\right))+2{\left(a^2{b}^2\right)}^{2/3}{\mathrm{A}\mathrm{i}}^{(1)}({(1/2\frac{\sqrt[3]{2}\sqrt[3]{a^2{b}^2}}{bx}+1/2\frac{ab\sqrt[3]{2}}{{\left(a^2{b}^2\right)}^{2/3}y\left(x\right)})}^2-1/2\frac{b{2}^{2/3}x}{\sqrt[3]{a^2{b}^2}})xy\left(x\right)){(a\mathrm{B}\mathrm{i}({(\frac{\sqrt[3]{2}}{2bx}\sqrt[3]{a^2{b}^2}+\frac{ab\sqrt[3]{2}}{2y\left(x\right)}{\left(a^2{b}^2\right)}^{-\frac{2}{3}})}^2-\frac{b{2}^{\frac{2}{3}}x}{2}\frac{1}{\sqrt[3]{a^2{b}^2}})b\sqrt[3]{2}(x+ay\left(x\right))+2{\left(a^2{b}^2\right)}^{2/3}{\mathrm{B}\mathrm{i}}^{(1)}({(1/2\frac{\sqrt[3]{2}\sqrt[3]{a^2{b}^2}}{bx}+1/2\frac{ab\sqrt[3]{2}}{{\left(a^2{b}^2\right)}^{2/3}y\left(x\right)})}^2-1/2\frac{b{2}^{2/3}x}{\sqrt[3]{a^2{b}^2}})xy\left(x\right))}^{-1}=0\}$$ 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