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 = 100.127 (sec), leaf count = 0 , could not solve
DSolve[a*y[x] + y[x]*Derivative[1][y][x] + Derivative[2][y][x] == y[x]^3, y[x], x]
Maple ✓
cpu = 1.344 (sec), leaf count = 1088
\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{-63\,{{\it \_a}}^{2}+63\,a} \left ( {\frac { \left ( -{\frac {1}{2}}+{\frac {i}{2}}\sqrt {3} \right ) ^{3}}{2} \left ( 126\,{\frac {1}{-{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3}}\sqrt [3]{-4\, \left ( {\it \_C1}\,\sqrt {5}\sqrt {{\frac {{\it \_C1}}{-{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3}}}}+1/4 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) ^{2} \left ( -{{\it \_a}}^{2}+a \right ) ^{3}}}-126\,{({{\it \_a}}^{6}-3\,a{{\it \_a}}^{4}+3\,{{\it \_a}}^{2}{a}^{2}-{a}^{3}){\frac {1}{\sqrt [3]{-4\, \left ( {\it \_C1}\,\sqrt {5}\sqrt {{\frac {{\it \_C1}}{-{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3}}}}+1/4 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) ^{2} \left ( -{{\it \_a}}^{2}+a \right ) ^{3}}}}}-126 \right ) }+63 \right ) }{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!{\frac {1}{ \left ( -{{\it \_a}}^{2}+a \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) } \left ( { \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) \left ( -{{\it \_a}}^{2}+a \right ) ^{3}{\frac {1}{\sqrt [3]{-4\, \left ( {\it \_C1}\,\sqrt {5}\sqrt {{\frac {{\it \_C1}}{-{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3}}}}+1/4 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) ^{2} \left ( -{{\it \_a}}^{2}+a \right ) ^{3}}}}}+\sqrt [3]{-4\, \left ( {\it \_C1}\,\sqrt {5}\sqrt {{\frac {{\it \_C1}}{-{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3}}}}+1/4 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) ^{2} \left ( -{{\it \_a}}^{2}+a \right ) ^{3}} \right ) }{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!{\frac {1}{ \left ( -2\,{{\it \_a}}^{2}+2\,a \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) } \left ( { \left ( i\sqrt {3}-1 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) \left ( -{{\it \_a}}^{2}+a \right ) ^{3}{\frac {1}{\sqrt [3]{-4\, \left ( {\it \_C1}\,\sqrt {5}\sqrt {{\frac {{\it \_C1}}{-{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3}}}}+1/4 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) ^{2} \left ( -{{\it \_a}}^{2}+a \right ) ^{3}}}}}+ \left ( -i\sqrt {3}-1 \right ) \sqrt [3]{-4\, \left ( {\it \_C1}\,\sqrt {5}\sqrt {{\frac {{\it \_C1}}{-{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3}}}}+1/4 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) ^{2} \left ( -{{\it \_a}}^{2}+a \right ) ^{3}} \right ) }{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!{\frac {1}{ \left ( -2\,{{\it \_a}}^{2}+2\,a \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) } \left ( -{ \left ( i\sqrt {3}+1 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) \left ( -{{\it \_a}}^{2}+a \right ) ^{3}{\frac {1}{\sqrt [3]{-4\, \left ( {\it \_C1}\,\sqrt {5}\sqrt {{\frac {{\it \_C1}}{-{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3}}}}+1/4 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) ^{2} \left ( -{{\it \_a}}^{2}+a \right ) ^{3}}}}}+\sqrt [3]{-4\, \left ( {\it \_C1}\,\sqrt {5}\sqrt {{\frac {{\it \_C1}}{-{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3}}}}+1/4 \right ) \left ( -{{\it \_a}}^{6}+3\,a{{\it \_a}}^{4}-3\,{{\it \_a}}^{2}{a}^{2}+80\,{{\it \_C1}}^{3}+{a}^{3} \right ) ^{2} \left ( -{{\it \_a}}^{2}+a \right ) ^{3}} \left ( i\sqrt {3}-1 \right ) \right ) }{d{\it \_a}}-x-{\it \_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
DSolve[a*y[x] + y[x]*Derivative[1][y][x] + Derivative[2][y][x] == y[x]^3, y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)+a*y(x) = y(x)^3, y(x),'implicit')
Maple raw output
Intat(((-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)*(-_a^2+a)^3/(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)+(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3))/(-_a^2+a)/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3),_a = y(x))-x-_C2 = 0, Intat(1/2*((I*3^(1/2)-1)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)*(-_a^2+a)^3/(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)+(-I*3^(1/2)-1)*(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3))/(-_a^2+a)/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3),_a = y(x))-x-_C2 = 0, Intat(1/2*(-(I*3^(1/2)+1)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)*(-_a^2+a)^3/(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)+(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)*(I*3^(1/2)-1))/(-_a^2+a)/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3),_a = y(x))-x-_C2 = 0, Intat(1/(-63*_a^2+63*a)*(1/2*(-1/2+1/2*I*3^(1/2))^3*(126/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)*(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)-126*(_a^6-3*_a^4*a+3*_a^2*a^2-a^3)/(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)-126)+63),_a = y(x))-x-_C2 = 0, Intat(1/2*((I*3^(1/2)-1)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)*(-_a^2+a)^3/(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)+(-I*3^(1/2)-1)*(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3))/(-_a^2+a)/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3),_a = y(x))-x-_C2 = 0, Intat(1/2*(-(I*3^(1/2)+1)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)*(-_a^2+a)^3/(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)+(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)*(I*3^(1/2)-1))/(-_a^2+a)/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3),_a = y(x))-x-_C2 = 0, Intat(((-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)*(-_a^2+a)^3/(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)+(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3))/(-_a^2+a)/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3),_a = y(x))-x-_C2 = 0, Intat(1/2*((I*3^(1/2)-1)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)*(-_a^2+a)^3/(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)+(-I*3^(1/2)-1)*(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3))/(-_a^2+a)/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3),_a = y(x))-x-_C2 = 0, Intat(1/2*(-(I*3^(1/2)+1)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)*(-_a^2+a)^3/(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)+(-4*(_C1*5^(1/2)*(_C1/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3))^(1/2)+1/4)*(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3)^2*(-_a^2+a)^3)^(1/3)*(I*3^(1/2)-1))/(-_a^2+a)/(-_a^6+3*_a^4*a-3*_a^2*a^2+80*_C1^3+a^3),_a = y(x))-x-_C2 = 0