4.6.25 $x^2{y}^{\prime }(x)=a{x}^{2}y(x{)}^2-ay(x{)}^3$

ODE
$$x^2{y}^{\prime }(x)=a{x}^{2}y(x{)}^2-ay(x{)}^3$$ ODE Classification

[_rational, _Abel]

Book solution method
Abel ODE, Second kind

Mathematica ✓
cpu = 0.617778 (sec), leaf count = 239

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

Maple ✓
cpu = 0.081 (sec), leaf count = 182

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