ODE
\[ y''(x)=a+b y(x)+2 y(x)^3 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.64971 (sec), leaf count = 570
\[\text {Solve}\left [\frac {4 \left (-y(x)+\text {Root}\left [\text {$\#$1}^4+\text {$\#$1}^2 b+2 \text {$\#$1} a+c_1\& ,1\right ]\right ) \left (-y(x)+\text {Root}\left [\text {$\#$1}^4+\text {$\#$1}^2 b+2 \text {$\#$1} a+c_1\& ,2\right ]\right ) \left (-y(x)+\text {Root}\left [\text {$\#$1}^4+\text {$\#$1}^2 b+2 \text {$\#$1} a+c_1\& ,3\right ]\right ) \left (-y(x)+\text {Root}\left [\text {$\#$1}^4+\text {$\#$1}^2 b+2 \text {$\#$1} a+c_1\& ,4\right ]\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,2\right ]-\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,4\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,1\right ]-y(x)\right )}{\left (\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,1\right ]-\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,4\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,2\right ]-y(x)\right )}}\right )|\frac {\left (\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,2\right ]-\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,1\right ]-\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,4\right ]\right )}{\left (\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,1\right ]-\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,2\right ]-\text {Root}\left [\text {$\#$1}^4+b \text {$\#$1}^2+2 a \text {$\#$1}+c_1\& ,4\right ]\right )}\right ){}^2}{\left (2 a y(x)+b y(x)^2+y(x)^4+c_1\right ) \left (\text {Root}\left [\text {$\#$1}^4+\text {$\#$1}^2 b+2 \text {$\#$1} a+c_1\& ,1\right ]-\text {Root}\left [\text {$\#$1}^4+\text {$\#$1}^2 b+2 \text {$\#$1} a+c_1\& ,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^4+\text {$\#$1}^2 b+2 \text {$\#$1} a+c_1\& ,2\right ]-\text {Root}\left [\text {$\#$1}^4+\text {$\#$1}^2 b+2 \text {$\#$1} a+c_1\& ,4\right ]\right )}=(x+c_2){}^2,y(x)\right ]\]
Maple ✓
cpu = 0.31 (sec), leaf count = 63
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{4}+\textit {\_a}^{2} b +2 a \textit {\_a} +\textit {\_C1}}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {\textit {\_a}^{4}+\textit {\_a}^{2} b +2 a \textit {\_a} +\textit {\_C1}}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[y''[x] == a + b*y[x] + 2*y[x]^3,y[x],x]
Mathematica raw output
Solve[(4*EllipticF[ArcSin[Sqrt[((Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 2] - Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 4])*(Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 1] - y[x]))/((Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 1] - Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 4])*(Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 2] - y[x]))]], ((Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 2] - Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 3])*(Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 1] - Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 4]))/((Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 1] - Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 3])*(Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 2] - Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 4]))]^2*(Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 1] - y[x])*(Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 2] - y[x])*(Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 3] - y[x])*(Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 4] - y[x]))/((Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 1] - Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 3])*(Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 2] - Root[C[1] + 2*a*#1 + b*#1^2 + #1^4 & , 4])*(C[1] + 2*a*y[x] + b*y[x]^2 + y[x]^4)) == (x + C[2])^2, y[x]]
Maple raw input
dsolve(diff(diff(y(x),x),x) = a+b*y(x)+2*y(x)^3, y(x))
Maple raw output
[Intat(1/(_a^4+_a^2*b+2*_a*a+_C1)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(_a^4+_a^2*b+2*_a*a+_C1)^(1/2),_a = y(x))-x-_C2 = 0]