ODE
\[ 2 y(x) y''(x)=6 y'(x)^2-y(x)^2 \left (a y(x)^3+1\right ) \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 22.4606 (sec), leaf count = 2761
\[\left \{\text {Solve}\left [\frac {4 \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-y(x)\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-y(x)\right )}}\right )|-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]+\Pi \left (\frac {\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-y(x)\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-y(x)\right )}}\right )|-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-y(x)\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-y(x)\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-y(x)\right )}} y(x) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-y(x)\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]-y(x)\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ){}^2 \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-y(x)\right ){}^2}} \sqrt {4 c_1 y(x)^6+4 a y(x)^5+y(x)^2}}=x+c_2,y(x)\right ],\text {Solve}\left [\frac {4 \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-y(x)\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-y(x)\right )}}\right )|-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]+\Pi \left (\frac {\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )};\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-y(x)\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-y(x)\right )}}\right )|-\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right )\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]-y(x)\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,3\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-y(x)\right )}} y(x) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]\right ) \left (y(x)-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right )}{\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ] \text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ] \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \sqrt {\frac {\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-y(x)\right ) \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]-y(x)\right )}{\left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,1\right ]-\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,4\right ]\right ){}^2 \left (\text {Root}\left [4 c_1 \text {$\#$1}^4+4 a \text {$\#$1}^3+1\& ,2\right ]-y(x)\right ){}^2}} \sqrt {4 c_1 y(x)^6+4 a y(x)^5+y(x)^2}}=x+c_2,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.084 (sec), leaf count = 71
\[ \left \{ \int ^{y \left ( x \right ) }\!-2\,{\frac {1}{\sqrt {4\,{\it \_C1}\,{{\it \_a}}^{4}+4\,{{\it \_a}}^{3}a+1}{\it \_a}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!2\,{\frac {1}{\sqrt {4\,{\it \_C1}\,{{\it \_a}}^{4}+4\,{{\it \_a}}^{3}a+1}{\it \_a}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[2*y[x]*y''[x] == -(y[x]^2*(1 + a*y[x]^3)) + 6*y'[x]^2,y[x],x]
Mathematica raw output
{Solve[(4*(EllipticF[ArcSin[Sqrt[((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - y[x]))/((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - y[x]))]], -(((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4]))/((-Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] + Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])))]*Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] + EllipticPi[(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2]*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4]))/(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1]*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])), ArcSin[Sqrt[((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - y[x]))/((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - y[x]))]], -(((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4]))/((-Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] + Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])))]*(-Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] + Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2]))*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - y[x])*Sqrt[((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3] - y[x]))/((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - y[x]))]*y[x]*(-Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4] + y[x]))/(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1]*Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2]*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*Sqrt[((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - y[x])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4] - y[x]))/((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])^2*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - y[x])^2)]*Sqrt[y[x]^2 + 4*a*y[x]^5 + 4*C[1]*y[x]^6]) == x + C[2], y[x]], Solve[(4*(EllipticF[ArcSin[Sqrt[((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - y[x]))/((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - y[x]))]], -(((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4]))/((-Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] + Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])))]*Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] + EllipticPi[(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2]*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4]))/(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1]*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])), ArcSin[Sqrt[((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - y[x]))/((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - y[x]))]], -(((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4]))/((-Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] + Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])))]*(-Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] + Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2]))*Sqrt[((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3] - y[x]))/((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 3])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - y[x]))]*y[x]*(-Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] + y[x])*(-Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4] + y[x]))/(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1]*Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2]*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*Sqrt[((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - y[x])*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4] - y[x]))/((Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 1] - Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 4])^2*(Root[1 + 4*a*#1^3 + 4*C[1]*#1^4 & , 2] - y[x])^2)]*Sqrt[y[x]^2 + 4*a*y[x]^5 + 4*C[1]*y[x]^6]) == x + C[2], y[x]]}
Maple raw input
dsolve(2*y(x)*diff(diff(y(x),x),x) = 6*diff(y(x),x)^2-(1+a*y(x)^3)*y(x)^2, y(x),'implicit')
Maple raw output
Intat(-2/(4*_C1*_a^4+4*_a^3*a+1)^(1/2)/_a,_a = y(x))-x-_C2 = 0, Intat(2/(4*_C1*_a^4+4*_a^3*a+1)^(1/2)/_a,_a = y(x))-x-_C2 = 0