ODE
\[ y(x) y''(x)=\text {a0}+\text {a1} y(x)+y(x)^3 (\text {a2}+\text {a3} y(x))+y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 5.28492 (sec), leaf count = 2633
\[\left \{\text {Solve}\left [-\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-y(x)\right )}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-y(x)\right )}}\right )|\frac {\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right )}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right )}\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-y(x)\right ){}^2 \sqrt {\frac {\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,3\right ]-y(x)\right )}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-y(x)\right )}} \sqrt {\frac {\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-y(x)\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]-y(x)\right )}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ){}^2 \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-y(x)\right ){}^2}}}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \sqrt {y(x) \left (y(x) \left (\text {a3} y(x)^2+2 \text {a2} y(x)+c_1\right )-2 \text {a1}\right )-\text {a0}}}=x+c_2,y(x)\right ],\text {Solve}\left [\frac {2 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-y(x)\right )}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-y(x)\right )}}\right )|\frac {\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right )}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right )}\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-y(x)\right ){}^2 \sqrt {\frac {\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,3\right ]-y(x)\right )}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,3\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-y(x)\right )}} \sqrt {\frac {\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-y(x)\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]-y(x)\right )}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ){}^2 \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-y(x)\right ){}^2}}}{\left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,1\right ]\right ) \left (\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,2\right ]-\text {Root}\left [\text {a3} \text {$\#$1}^4+2 \text {a2} \text {$\#$1}^3+c_1 \text {$\#$1}^2-2 \text {a1} \text {$\#$1}-\text {a0}\& ,4\right ]\right ) \sqrt {y(x) \left (y(x) \left (\text {a3} y(x)^2+2 \text {a2} y(x)+c_1\right )-2 \text {a1}\right )-\text {a0}}}=x+c_2,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.359 (sec), leaf count = 83
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{4} \mathit {a3} +2 \mathit {a2} \,\textit {\_a}^{3}+\textit {\_a}^{2} \textit {\_C1} -2 \mathit {a1} \textit {\_a} -\mathit {a0}}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {\textit {\_a}^{4} \mathit {a3} +2 \mathit {a2} \,\textit {\_a}^{3}+\textit {\_a}^{2} \textit {\_C1} -2 \mathit {a1} \textit {\_a} -\mathit {a0}}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[y[x]*y''[x] == a0 + a1*y[x] + y[x]^3*(a2 + a3*y[x]) + y'[x]^2,y[x],x]
Mathematica raw output
{Solve[(-2*EllipticF[ArcSin[Sqrt[((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - y[x]))/((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - y[x]))]], ((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 3])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4]))/((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 3])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4]))]*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - y[x])^2*Sqrt[((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 3] - y[x]))/((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 3])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - y[x]))]*Sqrt[((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - y[x])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4] - y[x]))/((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])^2*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - y[x])^2)])/((-Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] + Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*Sqrt[-a0 + y[x]*(-2*a1 + y[x]*(C[1] + 2*a2*y[x] + a3*y[x]^2))]) == x + C[2], y[x]], Solve[(2*EllipticF[ArcSin[Sqrt[((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - y[x]))/((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - y[x]))]], ((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 3])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4]))/((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 3])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4]))]*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - y[x])^2*Sqrt[((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 3] - y[x]))/((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 3])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - y[x]))]*Sqrt[((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - y[x])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4] - y[x]))/((Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])^2*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - y[x])^2)])/((-Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 1] + Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2])*(Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 2] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*Sqrt[-a0 + y[x]*(-2*a1 + y[x]*(C[1] + 2*a2*y[x] + a3*y[x]^2))]) == x + C[2], y[x]]}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+a0+a1*y(x)+(a2+a3*y(x))*y(x)^3, y(x))
Maple raw output
[Intat(1/(_a^4*a3+2*_a^3*a2+_C1*_a^2-2*_a*a1-a0)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(_a^4*a3+2*_a^3*a2+_C1*_a^2-2*_a*a1-a0)^(1/2),_a = y(x))-x-_C2 = 0]