ODE
\[ a y(x)+y''(x)+y(x) y'(x)=y(x)^3 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 10.5613 (sec), leaf count = 990
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\frac {e^{6 c_1} \left (a-K[1]^2\right )^2}{2 \sqrt [3]{e^{18 c_1} K[1]^6-3 a e^{18 c_1} K[1]^4+3 a^2 e^{18 c_1} K[1]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[1]^6+3 a e^{30 c_1} K[1]^4-3 a^2 e^{30 c_1} K[1]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}+\frac {1}{2} \left (a-K[1]^2\right )+\frac {1}{2} e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[1]^6-3 a e^{18 c_1} K[1]^4+3 a^2 e^{18 c_1} K[1]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[1]^6+3 a e^{30 c_1} K[1]^4-3 a^2 e^{30 c_1} K[1]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}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} \left (a-K[2]^2\right )^2}{4 \sqrt [3]{e^{18 c_1} K[2]^6-3 a e^{18 c_1} K[2]^4+3 a^2 e^{18 c_1} K[2]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[2]^6+3 a e^{30 c_1} K[2]^4-3 a^2 e^{30 c_1} K[2]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}+\frac {1}{2} \left (a-K[2]^2\right )-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[2]^6-3 a e^{18 c_1} K[2]^4+3 a^2 e^{18 c_1} K[2]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[2]^6+3 a e^{30 c_1} K[2]^4-3 a^2 e^{30 c_1} K[2]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}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} \left (a-K[3]^2\right )^2}{4 \sqrt [3]{e^{18 c_1} K[3]^6-3 a e^{18 c_1} K[3]^4+3 a^2 e^{18 c_1} K[3]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[3]^6+3 a e^{30 c_1} K[3]^4-3 a^2 e^{30 c_1} K[3]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}+\frac {1}{2} \left (a-K[3]^2\right )-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[3]^6-3 a e^{18 c_1} K[3]^4+3 a^2 e^{18 c_1} K[3]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[3]^6+3 a e^{30 c_1} K[3]^4-3 a^2 e^{30 c_1} K[3]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}dK[3]\& \right ][x+c_2]\right \}\right \}\]
Maple ✓
cpu = 4.089 (sec), leaf count = 108
\[\left [\int _{}^{y \left (x \right )}\frac {4 \RootOf \left (\left (-4 \textit {\_a}^{6}+12 a \,\textit {\_a}^{4}-12 \textit {\_a}^{2} a^{2}+4 a^{3}+320 \textit {\_C1} \right ) \textit {\_Z}^{9}+\left (-189 \textit {\_a}^{6}+567 a \,\textit {\_a}^{4}-567 \textit {\_a}^{2} a^{2}+189 a^{3}+15120 \textit {\_C1} \right ) \textit {\_Z}^{6}+238140 \textit {\_C1} \,\textit {\_Z}^{3}+1250235 \textit {\_C1} \right )^{3}+63}{-63 \textit {\_a}^{2}+63 a}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[a*y[x] + y[x]*y'[x] + y''[x] == y[x]^3,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Inactive[Integrate][((a - K[1]^2)/2 + (E^(6*C[1])*(a - K[1]^2)^2)/(2*(-2*E^(12*C[1]) - a^3*E^(18*C[1]) + 3*a^2*E^(18*C[1])*K[1]^2 - 3*a*E^(18*C[1])*K[1]^4 + E^(18*C[1])*K[1]^6 + 2*Sqrt[E^(24*C[1]) + a^3*E^(30*C[1]) - 3*a^2*E^(30*C[1])*K[1]^2 + 3*a*E^(30*C[1])*K[1]^4 - E^(30*C[1])*K[1]^6])^(1/3)) + (-2*E^(12*C[1]) - a^3*E^(18*C[1]) + 3*a^2*E^(18*C[1])*K[1]^2 - 3*a*E^(18*C[1])*K[1]^4 + E^(18*C[1])*K[1]^6 + 2*Sqrt[E^(24*C[1]) + a^3*E^(30*C[1]) - 3*a^2*E^(30*C[1])*K[1]^2 + 3*a*E^(30*C[1])*K[1]^4 - E^(30*C[1])*K[1]^6])^(1/3)/(2*E^(6*C[1])))^(-1), {K[1], 1, #1}] & ][x + C[2]]}, {y[x] -> InverseFunction[Inactive[Integrate][((a - K[2]^2)/2 - ((1 + I*Sqrt[3])*E^(6*C[1])*(a - K[2]^2)^2)/(4*(-2*E^(12*C[1]) - a^3*E^(18*C[1]) + 3*a^2*E^(18*C[1])*K[2]^2 - 3*a*E^(18*C[1])*K[2]^4 + E^(18*C[1])*K[2]^6 + 2*Sqrt[E^(24*C[1]) + a^3*E^(30*C[1]) - 3*a^2*E^(30*C[1])*K[2]^2 + 3*a*E^(30*C[1])*K[2]^4 - E^(30*C[1])*K[2]^6])^(1/3)) - ((1 - I*Sqrt[3])*(-2*E^(12*C[1]) - a^3*E^(18*C[1]) + 3*a^2*E^(18*C[1])*K[2]^2 - 3*a*E^(18*C[1])*K[2]^4 + E^(18*C[1])*K[2]^6 + 2*Sqrt[E^(24*C[1]) + a^3*E^(30*C[1]) - 3*a^2*E^(30*C[1])*K[2]^2 + 3*a*E^(30*C[1])*K[2]^4 - 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][((a - K[3]^2)/2 - ((1 - I*Sqrt[3])*E^(6*C[1])*(a - K[3]^2)^2)/(4*(-2*E^(12*C[1]) - a^3*E^(18*C[1]) + 3*a^2*E^(18*C[1])*K[3]^2 - 3*a*E^(18*C[1])*K[3]^4 + E^(18*C[1])*K[3]^6 + 2*Sqrt[E^(24*C[1]) + a^3*E^(30*C[1]) - 3*a^2*E^(30*C[1])*K[3]^2 + 3*a*E^(30*C[1])*K[3]^4 - E^(30*C[1])*K[3]^6])^(1/3)) - ((1 + I*Sqrt[3])*(-2*E^(12*C[1]) - a^3*E^(18*C[1]) + 3*a^2*E^(18*C[1])*K[3]^2 - 3*a*E^(18*C[1])*K[3]^4 + E^(18*C[1])*K[3]^6 + 2*Sqrt[E^(24*C[1]) + a^3*E^(30*C[1]) - 3*a^2*E^(30*C[1])*K[3]^2 + 3*a*E^(30*C[1])*K[3]^4 - 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)+a*y(x) = y(x)^3, y(x))
Maple raw output
[Intat(1/(-63*_a^2+63*a)*(4*RootOf((-4*_a^6+12*_a^4*a-12*_a^2*a^2+4*a^3+320*_C1)*_Z^9+(-189*_a^6+567*_a^4*a-567*_a^2*a^2+189*a^3+15120*_C1)*_Z^6+238140*_C1*_Z^3+1250235*_C1)^3+63),_a = y(x))-x-_C2 = 0]