ODE
\[ y''(x)+y(x) y'(x)=y(x)^3 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 4.07495 (sec), leaf count = 492
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2}{\frac {e^{6 c_1} K[1]^4}{\sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}-K[1]^2+e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}dK[1]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1+i \sqrt {3}\right ) e^{6 c_1} K[2]^4}{4 \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}-\frac {K[2]^2}{2}-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}dK[2]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1-i \sqrt {3}\right ) e^{6 c_1} K[3]^4}{4 \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}-\frac {K[3]^2}{2}-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}dK[3]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 0.282 (sec), leaf count = 336
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\frac {\left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {\textit {\_a}^{4}}{2 \left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {\textit {\_a}^{2}}{2}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}\frac {1}{-\frac {\left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {\textit {\_a}^{4}}{4 \left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {\textit {\_a}^{2}}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {\textit {\_a}^{4}}{2 \left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}\frac {1}{-\frac {\left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {\textit {\_a}^{4}}{4 \left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {\textit {\_a}^{2}}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {\textit {\_a}^{4}}{2 \left (\textit {\_a}^{6}+2 \textit {\_C1} +2 \sqrt {\textit {\_C1} \,\textit {\_a}^{6}+\textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[y[x]*y'[x] + y''[x] == y[x]^3,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Inactive[Integrate][2/(-K[1]^2 + (E^(6*C[1])*K[1]^4)/(-2*E^(12*C[1]) + E^(18*C[1])*K[1]^6 + 2*Sqrt[E^(24*C[1]) - E^(30*C[1])*K[1]^6])^(1/3) + (-2*E^(12*C[1]) + E^(18*C[1])*K[1]^6 + 2*Sqrt[E^(24*C[1]) - E^(30*C[1])*K[1]^6])^(1/3)/E^(6*C[1])), {K[1], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][(-1/2*K[2]^2 - ((1 + I*Sqrt[3])*E^(6*C[1])*K[2]^4)/(4*(-2*E^(12*C[1]) + E^(18*C[1])*K[2]^6 + 2*Sqrt[E^(24*C[1]) - E^(30*C[1])*K[2]^6])^(1/3)) - ((1 - I*Sqrt[3])*(-2*E^(12*C[1]) + E^(18*C[1])*K[2]^6 + 2*Sqrt[E^(24*C[1]) - E^(30*C[1])*K[2]^6])^(1/3))/(4*E^(6*C[1])))^(-1), {K[2], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][(-1/2*K[3]^2 - ((1 - I*Sqrt[3])*E^(6*C[1])*K[3]^4)/(4*(-2*E^(12*C[1]) + E^(18*C[1])*K[3]^6 + 2*Sqrt[E^(24*C[1]) - E^(30*C[1])*K[3]^6])^(1/3)) - ((1 + I*Sqrt[3])*(-2*E^(12*C[1]) + E^(18*C[1])*K[3]^6 + 2*Sqrt[E^(24*C[1]) - E^(30*C[1])*K[3]^6])^(1/3))/(4*E^(6*C[1])))^(-1), {K[3], 1, #1}] & ][x + C[2]]}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+y(x)*diff(y(x),x) = y(x)^3, y(x))
Maple raw output
[Intat(1/(1/2*(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3)+1/2*_a^4/(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3)-1/2*_a^2),_a = y(x))-x-_C2 = 0, Intat(1/(-1/4*(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3)-1/4*_a^4/(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3)-1/2*_a^2-1/2*I*3^(1/2)*(1/2*(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3)-1/2*_a^4/(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3))),_a = y(x))-x-_C2 = 0, Intat(1/(-1/4*(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3)-1/4*_a^4/(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3)-1/2*_a^2+1/2*I*3^(1/2)*(1/2*(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3)-1/2*_a^4/(_a^6+2*_C1+2*(_C1*_a^6+_C1^2)^(1/2))^(1/3))),_a = y(x))-x-_C2 = 0]