4.17.7 \(a x^2 y'(x)+b x y(x)+y'(x)^2=0\)

ODE
\[ a x^2 y'(x)+b x y(x)+y'(x)^2=0 \] ODE Classification

[[_homogeneous, `class G`]]

Book solution method
Homogeneous ODE, The Isobaric equation

Mathematica
cpu = 599.997 (sec), leaf count = 0 , timed out

$Aborted

Maple
cpu = 0.231 (sec), leaf count = 368

\[ \left \{ \int _{{\it \_b}}^{x}\!{1 \left (-{{\it \_a}}^{2}a-\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,by \relax (x ) } \right ) \left (a{{\it \_a}}^{3}+{\it \_a}\,\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,by \relax (x ) }+6\,y \relax (x ) \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \relax (x ) }\!-2\, \left (a{x}^{3}+x\sqrt {{a}^{2}{x}^{4}-4\,{\it \_f}\,bx}+6\,{\it \_f} \right ) ^{-1}-\int _{{\it \_b}}^{x}\!{1 \left (12\,{\frac {{\it \_a}\,{\it \_f}\,b}{\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,{\it \_f}\,b}}}+6\,{{\it \_a}}^{2}a+6\,\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,{\it \_f}\,b} \right ) \left (a{{\it \_a}}^{3}+{\it \_a}\,\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,{\it \_f}\,b}+6\,{\it \_f} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{1 \left (-{{\it \_a}}^{2}a+\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,by \relax (x ) } \right ) \left (a{{\it \_a}}^{3}-{\it \_a}\,\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,by \relax (x ) }+6\,y \relax (x ) \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \relax (x ) }\!2\, \left (-a{x}^{3}+x\sqrt {{a}^{2}{x}^{4}-4\,{\it \_f}\,bx}-6\,{\it \_f} \right ) ^{-1}-\int _{{\it \_b}}^{x}\!{1 \left (-12\,{\frac {{\it \_a}\,{\it \_f}\,b}{\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,{\it \_f}\,b}}}+6\,{{\it \_a}}^{2}a-6\,\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,{\it \_f}\,b} \right ) \left (a{{\it \_a}}^{3}-{\it \_a}\,\sqrt {{{\it \_a}}^{4}{a}^{2}-4\,{\it \_a}\,{\it \_f}\,b}+6\,{\it \_f} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[b*x*y[x] + a*x^2*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

$Aborted

Maple raw input

dsolve(diff(y(x),x)^2+a*x^2*diff(y(x),x)+b*x*y(x) = 0, y(x),'implicit')

Maple raw output

Int((-_a^2*a-(_a^4*a^2-4*_a*b*y(x))^(1/2))/(a*_a^3+_a*(_a^4*a^2-4*_a*b*y(x))^(1/
2)+6*y(x)),_a = _b .. x)+Intat(-2/(a*x^3+x*(a^2*x^4-4*_f*b*x)^(1/2)+6*_f)-Int((1
2*b*_a*_f/(_a^4*a^2-4*_a*_f*b)^(1/2)+6*_a^2*a+6*(_a^4*a^2-4*_a*_f*b)^(1/2))/(a*_
a^3+_a*(_a^4*a^2-4*_a*_f*b)^(1/2)+6*_f)^2,_a = _b .. x),_f = y(x))+_C1 = 0, Int(
(-_a^2*a+(_a^4*a^2-4*_a*b*y(x))^(1/2))/(a*_a^3-_a*(_a^4*a^2-4*_a*b*y(x))^(1/2)+6
*y(x)),_a = _b .. x)+Intat(2/(-a*x^3+x*(a^2*x^4-4*_f*b*x)^(1/2)-6*_f)-Int((-12*b
*_a*_f/(_a^4*a^2-4*_a*_f*b)^(1/2)+6*_a^2*a-6*(_a^4*a^2-4*_a*_f*b)^(1/2))/(a*_a^3
-_a*(_a^4*a^2-4*_a*_f*b)^(1/2)+6*_f)^2,_a = _b .. x),_f = y(x))+_C1 = 0