ODE
\[ y(x)^2 y''(x)=a \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.419748 (sec), leaf count = 65
\[\text {Solve}\left [\left (\frac {y(x) \sqrt {-\frac {2 a}{y(x)}+c_1}}{c_1}+\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {-\frac {2 a}{y(x)}+c_1}}{\sqrt {c_1}}\right )}{c_1{}^{3/2}}\right ){}^2=(x+c_2){}^2,y(x)\right ]\]
Maple ✓
cpu = 4.769 (sec), leaf count = 369
\[\left [y \left (x \right ) = \frac {\textit {\_C1} \left (a^{2} \textit {\_C1}^{2}+2 a \textit {\_C1} \,{\mathrm e}^{\RootOf \left (\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2}-2 \textit {\_Z} \,\textit {\_C1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C2} -2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) x \right )}+{\mathrm e}^{2 \RootOf \left (\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2}-2 \textit {\_Z} \,\textit {\_C1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C2} -2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) x \right )}\right ) {\mathrm e}^{-\RootOf \left (\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2}-2 \textit {\_Z} \,\textit {\_C1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{2}-2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C2} -2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) x \right )}}{2}, y \left (x \right ) = \frac {\textit {\_C1} \left (a^{2} \textit {\_C1}^{2}+2 a \textit {\_C1} \,{\mathrm e}^{\RootOf \left (\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2}-2 \textit {\_Z} \,\textit {\_C1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C2} +2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) x \right )}+{\mathrm e}^{2 \RootOf \left (\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2}-2 \textit {\_Z} \,\textit {\_C1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C2} +2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) x \right )}\right ) {\mathrm e}^{-\RootOf \left (\mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{4} a^{2}-2 \textit {\_Z} \,\textit {\_C1}^{3} a \,{\mathrm e}^{\textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C1}^{2}+2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) \textit {\_C2} +2 \,{\mathrm e}^{\textit {\_Z}} \mathrm {csgn}\left (\frac {1}{\textit {\_C1}}\right ) x \right )}}{2}\right ]\] Mathematica raw input
DSolve[y[x]^2*y''[x] == a,y[x],x]
Mathematica raw output
Solve[((2*a*ArcTanh[Sqrt[C[1] - (2*a)/y[x]]/Sqrt[C[1]]])/C[1]^(3/2) + (Sqrt[C[1] - (2*a)/y[x]]*y[x])/C[1])^2 == (x + C[2])^2, y[x]]
Maple raw input
dsolve(y(x)^2*diff(diff(y(x),x),x) = a, y(x))
Maple raw output
[y(x) = 1/2*_C1*(a^2*_C1^2+2*a*_C1*exp(RootOf(csgn(1/_C1)*_C1^4*a^2-2*_Z*_C1^3*a*exp(_Z)-exp(_Z)^2*csgn(1/_C1)*_C1^2-2*exp(_Z)*csgn(1/_C1)*_C2-2*exp(_Z)*csgn(1/_C1)*x))+exp(RootOf(csgn(1/_C1)*_C1^4*a^2-2*_Z*_C1^3*a*exp(_Z)-exp(_Z)^2*csgn(1/_C1)*_C1^2-2*exp(_Z)*csgn(1/_C1)*_C2-2*exp(_Z)*csgn(1/_C1)*x))^2)/exp(RootOf(csgn(1/_C1)*_C1^4*a^2-2*_Z*_C1^3*a*exp(_Z)-exp(_Z)^2*csgn(1/_C1)*_C1^2-2*exp(_Z)*csgn(1/_C1)*_C2-2*exp(_Z)*csgn(1/_C1)*x)), y(x) = 1/2*_C1*(a^2*_C1^2+2*a*_C1*exp(RootOf(csgn(1/_C1)*_C1^4*a^2-2*_Z*_C1^3*a*exp(_Z)-exp(_Z)^2*csgn(1/_C1)*_C1^2+2*exp(_Z)*csgn(1/_C1)*_C2+2*exp(_Z)*csgn(1/_C1)*x))+exp(RootOf(csgn(1/_C1)*_C1^4*a^2-2*_Z*_C1^3*a*exp(_Z)-exp(_Z)^2*csgn(1/_C1)*_C1^2+2*exp(_Z)*csgn(1/_C1)*_C2+2*exp(_Z)*csgn(1/_C1)*x))^2)/exp(RootOf(csgn(1/_C1)*_C1^4*a^2-2*_Z*_C1^3*a*exp(_Z)-exp(_Z)^2*csgn(1/_C1)*_C1^2+2*exp(_Z)*csgn(1/_C1)*_C2+2*exp(_Z)*csgn(1/_C1)*x))]