ODE
\[ 2 x^3 y'(x)^3+6 x^2 y(x) y'(x)^2-(1-6 x y(x)) y(x) y'(x)+2 y(x)^3=0 \] ODE Classification
[[_homogeneous, `class G`]]
Book solution method
Change of variable
Mathematica ✗
cpu = 600.001 (sec), leaf count = 0 , timed out
$Aborted
Maple ✓
cpu = 0.886 (sec), leaf count = 1532
\[ \left \{ \int _{{\it \_b}}^{x}\!{1 \left ( -{6}^{{\frac {2}{3}}} \left ( -9\,y \left ( x \right ) \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}+6\,{\it \_a}\,y \left ( x \right ) \left ( \sqrt [3]{-9\,y \left ( x \right ) \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) {{\it \_a}}^{2}}\sqrt [3]{6}-1 \right ) \right ) \left ( 6\,{{\it \_a}}^{2}y \left ( x \right ) +{\it \_a}\,{6}^{{\frac {2}{3}}} \left ( -9\,y \left ( x \right ) \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}} \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!6\,{x\sqrt [3]{6}\sqrt [3]{-9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f}} \left ( {6}^{2/3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}+6\,{\it \_f}\,x \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!324\,{{\it \_a}\,\sqrt [3]{6}{\it \_f} \left ( {{\it \_a}\,{\it \_f} \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}-2/3\,{\it \_a}\, \left ( -1/18\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{-2/3}}-1/6\, \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}+1/3\,{\it \_a}\, \left ( -2/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) {6}^{2/3} \right ) \left ( {6}^{2/3} \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}+6\,{\it \_a}\,{\it \_f} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{\frac {1}{6\,{\it \_a}} \left ( {6}^{{\frac {2}{3}}} \left ( i\sqrt {3}-1 \right ) \left ( -9\,y \left ( x \right ) \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}-6\, \left ( i\sqrt {3}+2\,\sqrt [3]{-9\,y \left ( x \right ) \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) {{\it \_a}}^{2}}\sqrt [3]{6}+1 \right ) y \left ( x \right ) {\it \_a} \right ) \left ( -{\frac { \left ( i\sqrt {3}-1 \right ) {6}^{{\frac {2}{3}}}}{6} \left ( -9\,y \left ( x \right ) \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}}+ \left ( i\sqrt {3}+1 \right ) y \left ( x \right ) {\it \_a} \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!12\,{x\sqrt [3]{6}\sqrt [3]{-9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f}} \left ( i{6}^{2/3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}\sqrt {3}-{6}^{2/3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}-6\,i\sqrt {3}x{\it \_f}-6\,{\it \_f}\,x \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!-18\,{{\it \_a}\,\sqrt [3]{6}{\it \_f} \left ( { \left ( i\sqrt {3}+1 \right ) {\it \_a}\,{\it \_f} \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}-2/3\,{\it \_a}\, \left ( -1/18\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{-2/3}}+1/6\, \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}+1/3\,{\it \_a}\, \left ( -2/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) \left ( i\sqrt {3}-1 \right ) {6}^{2/3} \right ) \left ( -1/6\, \left ( i\sqrt {3}-1 \right ) {6}^{2/3} \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}+ \left ( i\sqrt {3}+1 \right ) {\it \_a}\,{\it \_f} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{\frac {1}{6\,{\it \_a}} \left ( {6}^{{\frac {2}{3}}} \left ( i\sqrt {3}+1 \right ) \left ( -9\,y \left ( x \right ) \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}-6\, \left ( i\sqrt {3}-2\,\sqrt [3]{-9\,y \left ( x \right ) \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) {{\it \_a}}^{2}}\sqrt [3]{6}-1 \right ) y \left ( x \right ) {\it \_a} \right ) \left ( -{\frac { \left ( i\sqrt {3}+1 \right ) {6}^{{\frac {2}{3}}}}{6} \left ( -9\,y \left ( x \right ) \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\, \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) }{{\it \_a}}}}+y \left ( x \right ) \right ) {{\it \_a}}^{2} \right ) ^{{\frac {2}{3}}}}+ \left ( i\sqrt {3}-1 \right ) y \left ( x \right ) {\it \_a} \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!-12\,{x\sqrt [3]{6}\sqrt [3]{-9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f}} \left ( i{6}^{2/3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}\sqrt {3}+{6}^{2/3} \left ( -9\,{x}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{{\it \_f}}^{2}x-2\,{\it \_f}}{x}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}-6\,i\sqrt {3}x{\it \_f}+6\,{\it \_f}\,x \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!18\,{{\it \_a}\,\sqrt [3]{6}{\it \_f} \left ( {{\it \_a}\, \left ( i\sqrt {3}-1 \right ) {\it \_f} \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}-2/3\,{\it \_a}\, \left ( -1/18\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{-2/3}}+1/6\, \left ( i\sqrt {3}+1 \right ) \left ( {{\it \_f}\,\sqrt {3} \left ( {\it \_a}\,{\it \_f}-1/27 \right ) {\frac {1}{\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}}}}+1/3\,{\it \_a}\, \left ( -2/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) \right ) {6}^{2/3} \right ) \left ( -1/6\, \left ( i\sqrt {3}+1 \right ) {6}^{2/3} \left ( -9\,{{\it \_a}}^{2} \left ( -1/9\,\sqrt {3}\sqrt {{\frac {27\,{\it \_a}\,{{\it \_f}}^{2}-2\,{\it \_f}}{{\it \_a}}}}+{\it \_f} \right ) {\it \_f} \right ) ^{2/3}+{\it \_a}\, \left ( i\sqrt {3}-1 \right ) {\it \_f} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[2*y[x]^3 - y[x]*(1 - 6*x*y[x])*y'[x] + 6*x^2*y[x]*y'[x]^2 + 2*x^3*y'[x]^3 == 0,y[x],x]
Mathematica raw output
$Aborted
Maple raw input
dsolve(2*x^3*diff(y(x),x)^3+6*x^2*y(x)*diff(y(x),x)^2-(1-6*x*y(x))*y(x)*diff(y(x),x)+2*y(x)^3 = 0, y(x),'implicit')
Maple raw output
Int((-6^(2/3)*(-9*y(x)*(-1/9*3^(1/2)*((27*_a*y(x)^2-2*y(x))/_a)^(1/2)+y(x))*_a^2)^(2/3)+6*_a*y(x)*((-9*y(x)*(-1/9*3^(1/2)*((27*_a*y(x)^2-2*y(x))/_a)^(1/2)+y(x))*_a^2)^(1/3)*6^(1/3)-1))/(6*_a^2*y(x)+_a*6^(2/3)*(-9*y(x)*(-1/9*3^(1/2)*((27*_a*y(x)^2-2*y(x))/_a)^(1/2)+y(x))*_a^2)^(2/3)),_a = _b .. x)+Intat(6*x*6^(1/3)*(-9*x^2*(-1/9*3^(1/2)*((27*_f^2*x-2*_f)/x)^(1/2)+_f)*_f)^(1/3)/(6^(2/3)*(-9*x^2*(-1/9*3^(1/2)*((27*_f^2*x-2*_f)/x)^(1/2)+_f)*_f)^(2/3)+6*_f*x)-Int(324*_a*6^(1/3)*_f*(_a*(_f*3^(1/2)*(_a*_f-1/27)/((27*_a*_f^2-2*_f)/_a)^(1/2)-2/3*_a*(-1/18*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f))*_f/(-9*_a^2*(-1/9*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f)*_f)^(2/3)-1/6*(_f*3^(1/2)*(_a*_f-1/27)/((27*_a*_f^2-2*_f)/_a)^(1/2)+1/3*_a*(-2/9*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f))*6^(2/3))/(6^(2/3)*(-9*_a^2*(-1/9*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f)*_f)^(2/3)+6*_a*_f)^2,_a = _b .. x),_f = y(x))+_C1 = 0, Int(1/6*(6^(2/3)*(I*3^(1/2)+1)*(-9*y(x)*(-1/9*3^(1/2)*((27*_a*y(x)^2-2*y(x))/_a)^(1/2)+y(x))*_a^2)^(2/3)-6*(I*3^(1/2)-2*(-9*y(x)*(-1/9*3^(1/2)*((27*_a*y(x)^2-2*y(x))/_a)^(1/2)+y(x))*_a^2)^(1/3)*6^(1/3)-1)*y(x)*_a)/(-1/6*6^(2/3)*(I*3^(1/2)+1)*(-9*y(x)*(-1/9*3^(1/2)*((27*_a*y(x)^2-2*y(x))/_a)^(1/2)+y(x))*_a^2)^(2/3)+(I*3^(1/2)-1)*y(x)*_a)/_a,_a = _b .. x)+Intat(-12*x*6^(1/3)*(-9*x^2*(-1/9*3^(1/2)*((27*_f^2*x-2*_f)/x)^(1/2)+_f)*_f)^(1/3)/(I*6^(2/3)*(-9*x^2*(-1/9*3^(1/2)*((27*_f^2*x-2*_f)/x)^(1/2)+_f)*_f)^(2/3)*3^(1/2)+6^(2/3)*(-9*x^2*(-1/9*3^(1/2)*((27*_f^2*x-2*_f)/x)^(1/2)+_f)*_f)^(2/3)-6*I*3^(1/2)*x*_f+6*_f*x)-Int(18*(_a*(_f*3^(1/2)*(_a*_f-1/27)/((27*_a*_f^2-2*_f)/_a)^(1/2)-2/3*_a*(-1/18*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f))*(I*3^(1/2)-1)*_f/(-9*_a^2*(-1/9*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f)*_f)^(2/3)+1/6*(I*3^(1/2)+1)*(_f*3^(1/2)*(_a*_f-1/27)/((27*_a*_f^2-2*_f)/_a)^(1/2)+1/3*_a*(-2/9*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f))*6^(2/3))*_a*6^(1/3)*_f/(-1/6*(I*3^(1/2)+1)*6^(2/3)*(-9*_a^2*(-1/9*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f)*_f)^(2/3)+_a*(I*3^(1/2)-1)*_f)^2,_a = _b .. x),_f = y(x))+_C1 = 0, Int(1/6*(6^(2/3)*(I*3^(1/2)-1)*(-9*y(x)*(-1/9*3^(1/2)*((27*_a*y(x)^2-2*y(x))/_a)^(1/2)+y(x))*_a^2)^(2/3)-6*(I*3^(1/2)+2*(-9*y(x)*(-1/9*3^(1/2)*((27*_a*y(x)^2-2*y(x))/_a)^(1/2)+y(x))*_a^2)^(1/3)*6^(1/3)+1)*y(x)*_a)/_a/(-1/6*6^(2/3)*(I*3^(1/2)-1)*(-9*y(x)*(-1/9*3^(1/2)*((27*_a*y(x)^2-2*y(x))/_a)^(1/2)+y(x))*_a^2)^(2/3)+(I*3^(1/2)+1)*y(x)*_a),_a = _b .. x)+Intat(12*x*6^(1/3)*(-9*x^2*(-1/9*3^(1/2)*((27*_f^2*x-2*_f)/x)^(1/2)+_f)*_f)^(1/3)/(I*6^(2/3)*(-9*x^2*(-1/9*3^(1/2)*((27*_f^2*x-2*_f)/x)^(1/2)+_f)*_f)^(2/3)*3^(1/2)-6^(2/3)*(-9*x^2*(-1/9*3^(1/2)*((27*_f^2*x-2*_f)/x)^(1/2)+_f)*_f)^(2/3)-6*I*3^(1/2)*x*_f-6*_f*x)-Int(-18*_a*6^(1/3)*_f*((I*3^(1/2)+1)*_a*(_f*3^(1/2)*(_a*_f-1/27)/((27*_a*_f^2-2*_f)/_a)^(1/2)-2/3*_a*(-1/18*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f))*_f/(-9*_a^2*(-1/9*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f)*_f)^(2/3)+1/6*(_f*3^(1/2)*(_a*_f-1/27)/((27*_a*_f^2-2*_f)/_a)^(1/2)+1/3*_a*(-2/9*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f))*(I*3^(1/2)-1)*6^(2/3))/(-1/6*(I*3^(1/2)-1)*6^(2/3)*(-9*_a^2*(-1/9*3^(1/2)*((27*_a*_f^2-2*_f)/_a)^(1/2)+_f)*_f)^(2/3)+(I*3^(1/2)+1)*_a*_f)^2,_a = _b .. x),_f = y(x))+_C1 = 0