ODE
\[ y''(x)=\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.62597 (sec), leaf count = 869
\[\text {Solve}\left [\frac {24 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,2\right ]-\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,4\right ]\right ) \left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,1\right ]-y(x)\right )}{\left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,1\right ]-\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,4\right ]\right ) \left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,2\right ]-y(x)\right )}}\right )|\frac {\left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,2\right ]-\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,3\right ]\right ) \left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,1\right ]-\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,4\right ]\right )}{\left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,1\right ]-\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,3\right ]\right ) \left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,2\right ]-\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,4\right ]\right )}\right ){}^2 \left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,1\right ]-y(x)\right ) \left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,2\right ]-y(x)\right ) \left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,3\right ]-y(x)\right ) \left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,4\right ]-y(x)\right )}{\left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,1\right ]-\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,3\right ]\right ) \left (\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,2\right ]-\text {Root}\left [3 \text {a3} \text {$\#$1}^4+4 \text {a2} \text {$\#$1}^3+6 \text {a1} \text {$\#$1}^2+12 \text {a0} \text {$\#$1}+6 c_1\& ,4\right ]\right ) \left (3 \text {a3} y(x)^4+4 \text {a2} y(x)^3+6 \text {a1} y(x)^2+12 \text {a0} y(x)+6 c_1\right )}=\left (x+c_2\right ){}^2,y(x)\right ]\]
Maple ✓
cpu = 0.087 (sec), leaf count = 89
\[ \left \{ \int ^{y \left ( x \right ) }\!-6\,{\frac {1}{\sqrt {18\,{\it a3}\,{{\it \_a}}^{4}+24\,{\it a2}\,{{\it \_a}}^{3}+36\,{\it a1}\,{{\it \_a}}^{2}+72\,{\it a0}\,{\it \_a}+36\,{\it \_C1}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!6\,{\frac {1}{\sqrt {18\,{\it a3}\,{{\it \_a}}^{4}+24\,{\it a2}\,{{\it \_a}}^{3}+36\,{\it a1}\,{{\it \_a}}^{2}+72\,{\it a0}\,{\it \_a}+36\,{\it \_C1}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y''[x] == a0 + a1*y[x] + a2*y[x]^2 + a3*y[x]^3,y[x],x]
Mathematica raw output
Solve[(24*EllipticF[ArcSin[Sqrt[((Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 2] - Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 4])*(Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 1] - y[x]))/((Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 1] - Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 4])*(Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 2] - y[x]))]], ((Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 2] - Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 3])*(Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 1] - Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 4]))/((Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 1] - Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 3])*(Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 2] - Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 4]))]^2*(Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 1] - y[x])*(Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 2] - y[x])*(Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 3] - y[x])*(Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 4] - y[x]))/((Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 1] - Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 3])*(Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 2] - Root[6*C[1] + 12*a0*#1 + 6*a1*#1^2 + 4*a2*#1^3 + 3*a3*#1^4 & , 4])*(6*C[1] + 12*a0*y[x] + 6*a1*y[x]^2 + 4*a2*y[x]^3 + 3*a3*y[x]^4)) == (x + C[2])^2, y[x]]
Maple raw input
dsolve(diff(diff(y(x),x),x) = a0+a1*y(x)+a2*y(x)^2+a3*y(x)^3, y(x),'implicit')
Maple raw output
Intat(-6/(18*_a^4*a3+24*_a^3*a2+36*_a^2*a1+72*_a*a0+36*_C1)^(1/2),_a = y(x))-x-_C2 = 0, Intat(6/(18*_a^4*a3+24*_a^3*a2+36*_a^2*a1+72*_a*a0+36*_C1)^(1/2),_a = y(x))-x-_C2 = 0