4.39.1 \(y(x) y''(x)=\text {a0}+\text {a1} y(x)+y(x)^3 (\text {a2}+\text {a3} y(x))+y'(x)^2\)

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 = 4.93952 (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 = 0.125 (sec), leaf count = 83

\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {{\it a3}\,{{\it \_a}}^{4}+2\,{{\it \_a}}^{3}{\it a2}+{{\it \_a}}^{2}{\it \_C1}-2\,{\it a1}\,{\it \_a}-{\it a0}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{\sqrt {{\it a3}\,{{\it \_a}}^{4}+2\,{{\it \_a}}^{3}{\it a2}+{{\it \_a}}^{2}{\it \_C1}-2\,{\it a1}\,{\it \_a}-{\it a0}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] 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])*(R
oot[-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 + a
3*#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*a
2*#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]))]*Sq
rt[((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[-a
0 - 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])*(R
oot[-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[((R
oot[-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 + a
3*#1^4 & , 1] - Root[-a0 - 2*a1*#1 + C[1]*#1^2 + 2*a2*#1^3 + a3*#1^4 & , 4])*(Ro
ot[-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)])/((-R
oot[-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),'implicit')

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, Inta
t(-1/(_a^4*a3+2*_a^3*a2+_C1*_a^2-2*_a*a1-a0)^(1/2),_a = y(x))-x-_C2 = 0