ODE
\[ x y(x)^2 y''(x)=a \] ODE Classification
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.438511 (sec), leaf count = 103
\[\text {Solve}\left [2 \sqrt {2} \sqrt {-\frac {y(x) (a x+c_1 y(x))}{x^2}}+\frac {\sqrt {2} a \tan ^{-1}\left (\frac {a x+2 c_1 y(x)}{2 \sqrt {c_1} x \sqrt {-\frac {y(x) (a x+c_1 y(x))}{x^2}}}\right )}{\sqrt {c_1}}+\frac {4 c_1}{x}+4 c_1 c_2=0,y(x)\right ]\]
Maple ✓
cpu = 10.094 (sec), leaf count = 793
\[\left [y \left (x \right ) = \frac {\textit {\_C1} x \left (81 a^{2} \textit {\_C1}^{2}+18 a \textit {\_C1} \,{\mathrm e}^{\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x -2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x -2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x -2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2}, y \left (x \right ) = \frac {\textit {\_C1} x \left (81 a^{2} \textit {\_C1}^{2}+18 a \textit {\_C1} \,{\mathrm e}^{\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x +2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x +2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x +2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2}, y \left (x \right ) = \frac {\textit {\_C1} x \left (81 a^{2} \textit {\_C1}^{2}+18 a \textit {\_C1} \,{\mathrm e}^{\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x +2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x +2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x -54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x +6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x +2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2}, y \left (x \right ) = \frac {\textit {\_C1} x \left (81 a^{2} \textit {\_C1}^{2}+18 a \textit {\_C1} \,{\mathrm e}^{\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x -2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}+{\mathrm e}^{2 \RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x -2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\RootOf \left (243 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2} x +54 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} a x \,\textit {\_C1}^{3}-3 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{2 \textit {\_Z}} \textit {\_C1}^{2} x -6 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}} \textit {\_C2} x -2 \,\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) {\mathrm e}^{\textit {\_Z}}\right )}}{2}\right ]\] Mathematica raw input
DSolve[x*y[x]^2*y''[x] == a,y[x],x]
Mathematica raw output
Solve[(Sqrt[2]*a*ArcTan[(a*x + 2*C[1]*y[x])/(2*x*Sqrt[C[1]]*Sqrt[-((y[x]*(a*x + C[1]*y[x]))/x^2)])])/Sqrt[C[1]] + (4*C[1])/x + 4*C[1]*C[2] + 2*Sqrt[2]*Sqrt[-((y[x]*(a*x + C[1]*y[x]))/x^2)] == 0, y[x]]
Maple raw input
dsolve(x*y(x)^2*diff(diff(y(x),x),x) = a, y(x))
Maple raw output
[y(x) = 1/2*_C1*x*(81*a^2*_C1^2+18*a*_C1*exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x-54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x-6*csgn(1/_C1)*exp(_Z)*_C2*x-2*csgn(1/_C1)*exp(_Z)))+exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x-54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x-6*csgn(1/_C1)*exp(_Z)*_C2*x-2*csgn(1/_C1)*exp(_Z)))^2)/exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x-54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x-6*csgn(1/_C1)*exp(_Z)*_C2*x-2*csgn(1/_C1)*exp(_Z))), y(x) = 1/2*_C1*x*(81*a^2*_C1^2+18*a*_C1*exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x+54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x+6*csgn(1/_C1)*exp(_Z)*_C2*x+2*csgn(1/_C1)*exp(_Z)))+exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x+54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x+6*csgn(1/_C1)*exp(_Z)*_C2*x+2*csgn(1/_C1)*exp(_Z)))^2)/exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x+54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x+6*csgn(1/_C1)*exp(_Z)*_C2*x+2*csgn(1/_C1)*exp(_Z))), y(x) = 1/2*_C1*x*(81*a^2*_C1^2+18*a*_C1*exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x-54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x+6*csgn(1/_C1)*exp(_Z)*_C2*x+2*csgn(1/_C1)*exp(_Z)))+exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x-54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x+6*csgn(1/_C1)*exp(_Z)*_C2*x+2*csgn(1/_C1)*exp(_Z)))^2)/exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x-54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x+6*csgn(1/_C1)*exp(_Z)*_C2*x+2*csgn(1/_C1)*exp(_Z))), y(x) = 1/2*_C1*x*(81*a^2*_C1^2+18*a*_C1*exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x+54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x-6*csgn(1/_C1)*exp(_Z)*_C2*x-2*csgn(1/_C1)*exp(_Z)))+exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x+54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x-6*csgn(1/_C1)*exp(_Z)*_C2*x-2*csgn(1/_C1)*exp(_Z)))^2)/exp(RootOf(243*csgn(1/_C1)*_C1^4*a^2*x+54*_Z*exp(_Z)*a*x*_C1^3-3*csgn(1/_C1)*exp(_Z)^2*_C1^2*x-6*csgn(1/_C1)*exp(_Z)*_C2*x-2*csgn(1/_C1)*exp(_Z)))]