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 = 9.46842 (sec), leaf count = 113
\[\left \{\left \{y(x)\to \frac {\int _1^x\frac {\text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}^2-8 \text {$\#$1}^3} \tan ^{-1}\left (\sqrt {8 \text {$\#$1}-1}\right )}{\text {$\#$1} \sqrt {8 \text {$\#$1}-1}}+\log (8 \text {$\#$1}-1)+\log \left (1+\frac {1}{8 \text {$\#$1}-1}\right )+\frac {3 \sqrt {\text {$\#$1}^2-8 \text {$\#$1}^3}}{\text {$\#$1}}\& \right ][c_1+2 \log (K[1])]}{K[1]}dK[1]}{x}\right \}\right \}\]
Maple ✓
cpu = 2.131 (sec), leaf count = 3181
\[\text {Expression too large to display}\] 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
{{y[x] -> Inactive[Integrate][InverseFunction[Log[-1 + 8*#1] + Log[1 + (-1 + 8*#1)^(-1)] + (3*Sqrt[#1^2 - 8*#1^3])/#1 - (2*ArcTan[Sqrt[-1 + 8*#1]]*Sqrt[#1^2 - 8*#1^3])/(#1*Sqrt[-1 + 8*#1]) & ][C[1] + 2*Log[K[1]]]/K[1], {K[1], 1, x}]/x}}
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))
Maple raw output
[Int(-(6^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)-6*y(x)*_a*6^(1/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(1/3)+6*_a*y(x))/_a/(6^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)+6*_a*y(x)),_a = _b .. x)+Intat(6*x*6^(1/3)*(_f*(3^(1/2)*(_f*(27*_f*x-2)/x)^(1/2)-9*_f)*x^2)^(1/3)/(6^(2/3)*(_f*(3^(1/2)*(_f*(27*_f*x-2)/x)^(1/2)-9*_f)*x^2)^(2/3)+6*_f*x)-Int(-(2/3*6^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)-6*_a*6^(1/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)-2*_f*_a*6^(1/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)+6*_a)/_a/(6^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)+6*_a*_f)+(6^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-6*_f*_a*6^(1/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)+6*_a*_f)/_a/(6^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)+6*_a*_f)^2*(2/3*6^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)+6*_a),_a = _b .. x),_f = y(x))+_C1 = 0, Int(-(3*I*3^(1/6)*2^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)-6*I*3^(1/2)*_a*y(x)+6^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)+12*y(x)*_a*6^(1/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(1/3)+6*_a*y(x))/_a/(3*I*3^(1/6)*2^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)-6*I*3^(1/2)*_a*y(x)+6^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)+6*_a*y(x)),_a = _b .. x)+Intat(-12*x*6^(1/3)*(_f*(3^(1/2)*(_f*(27*_f*x-2)/x)^(1/2)-9*_f)*x^2)^(1/3)/(I*6^(2/3)*(_f*(3^(1/2)*(_f*(27*_f*x-2)/x)^(1/2)-9*_f)*x^2)^(2/3)*3^(1/2)-6*I*3^(1/2)*x*_f+6^(2/3)*(_f*(3^(1/2)*(_f*(27*_f*x-2)/x)^(1/2)-9*_f)*x^2)^(2/3)+6*_f*x)-Int(-(2*I*3^(1/6)*2^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)-6*I*3^(1/2)*_a+2/3*6^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)+12*_a*6^(1/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)+4*_f*_a*6^(1/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)+6*_a)/_a/(3*I*3^(1/6)*2^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-6*I*3^(1/2)*_a*_f+6^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)+6*_a*_f)+(3*I*3^(1/6)*2^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-6*I*3^(1/2)*_a*_f+6^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)+12*_f*_a*6^(1/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)+6*_a*_f)/_a/(3*I*3^(1/6)*2^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-6*I*3^(1/2)*_a*_f+6^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)+6*_a*_f)^2*(2*I*3^(1/6)*2^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)-6*I*3^(1/2)*_a+2/3*6^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)+6*_a),_a = _b .. x),_f = y(x))+_C1 = 0, Int(-(3*I*3^(1/6)*2^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)-6*I*3^(1/2)*_a*y(x)-6^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)-12*y(x)*_a*6^(1/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(1/3)-6*_a*y(x))/_a/(3*I*3^(1/6)*2^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)-6*I*3^(1/2)*_a*y(x)-6^(2/3)*(y(x)*(3^(1/2)*(y(x)*(27*_a*y(x)-2)/_a)^(1/2)-9*y(x))*_a^2)^(2/3)-6*_a*y(x)),_a = _b .. x)+Intat(12*x*6^(1/3)*(_f*(3^(1/2)*(_f*(27*_f*x-2)/x)^(1/2)-9*_f)*x^2)^(1/3)/(I*6^(2/3)*(_f*(3^(1/2)*(_f*(27*_f*x-2)/x)^(1/2)-9*_f)*x^2)^(2/3)*3^(1/2)-6*I*3^(1/2)*x*_f-6^(2/3)*(_f*(3^(1/2)*(_f*(27*_f*x-2)/x)^(1/2)-9*_f)*x^2)^(2/3)-6*_f*x)-Int(-(2*I*3^(1/6)*2^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)-6*I*3^(1/2)*_a-2/3*6^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)-12*_a*6^(1/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)-4*_f*_a*6^(1/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)-6*_a)/_a/(3*I*3^(1/6)*2^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-6*I*3^(1/2)*_a*_f-6^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-6*_a*_f)+(3*I*3^(1/6)*2^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-6*I*3^(1/2)*_a*_f-6^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-12*_f*_a*6^(1/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)-6*_a*_f)/_a/(3*I*3^(1/6)*2^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-6*I*3^(1/2)*_a*_f-6^(2/3)*(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(2/3)-6*_a*_f)^2*(2*I*3^(1/6)*2^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)-6*I*3^(1/2)*_a-2/3*6^(2/3)/(_f*(3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2)^(1/3)*((3^(1/2)*(_f*(27*_a*_f-2)/_a)^(1/2)-9*_f)*_a^2+_f*(1/2*3^(1/2)/(_f*(27*_a*_f-2)/_a)^(1/2)*((27*_a*_f-2)/_a+27*_f)-9)*_a^2)-6*_a),_a = _b .. x),_f = y(x))+_C1 = 0]