ODE
\[ x y(x)^2 y''(x)=a \] ODE Classification
[[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.322733 (sec), leaf count = 103
\[\text {Solve}\left [2 \sqrt {2} \sqrt {-\frac {y(x) \left (a x+c_1 y(x)\right )}{x^2}}+\frac {\sqrt {2} a \tan ^{-1}\left (\frac {a x+2 c_1 y(x)}{2 \sqrt {c_1} x \sqrt {-\frac {y(x) \left (a x+c_1 y(x)\right )}{x^2}}}\right )}{\sqrt {c_1}}+\frac {4 c_1}{x}+4 c_1 c_2=0,y(x)\right ]\]
Maple ✓
cpu = 0.268 (sec), leaf count = 209
\[ \left \{ {\frac {1}{x} \left ( 9\,a\ln \left ( \left ( {\frac {1}{{{\it \_C1}}^{2}} \left ( {\frac {y \left ( x \right ) }{x}}-18\,a{{\it \_C1}}^{2} \right ) }+9\,a \right ) {\it \_C1}+\sqrt {{\frac {-18\,a{{\it \_C1}}^{2}xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}{{\it \_C1}}^{2}}}} \right ) x+ \left ( {{\it \_C1}}^{-2} \right ) ^{{\frac {3}{2}}} \left ( {{\it \_C1}}^{2}\sqrt {{\frac {-18\,a{{\it \_C1}}^{2}xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}{{\it \_C1}}^{2}}}}x-{\it \_C2}\,x-{\frac {1}{3}} \right ) \right ) \left ( {{\it \_C1}}^{-2} \right ) ^{-{\frac {3}{2}}}}=0,-{\frac {1}{x} \left ( 9\,a\ln \left ( \left ( {\frac {1}{{{\it \_C1}}^{2}} \left ( {\frac {y \left ( x \right ) }{x}}-18\,a{{\it \_C1}}^{2} \right ) }+9\,a \right ) {\it \_C1}+\sqrt {{\frac {-18\,a{{\it \_C1}}^{2}xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}{{\it \_C1}}^{2}}}} \right ) x+ \left ( {{\it \_C1}}^{-2} \right ) ^{{\frac {3}{2}}} \left ( {{\it \_C1}}^{2}\sqrt {{\frac {-18\,a{{\it \_C1}}^{2}xy \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}{{\it \_C1}}^{2}}}}x+{\it \_C2}\,x+{\frac {1}{3}} \right ) \right ) \left ( {{\it \_C1}}^{-2} \right ) ^{-{\frac {3}{2}}}}=0 \right \} \] 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),'implicit')
Maple raw output
-1/(1/_C1^2)^(3/2)*(9*a*ln(((y(x)/x-18*a*_C1^2)/_C1^2+9*a)*_C1+((-18*a*_C1^2*x*y(x)+y(x)^2)/x^2/_C1^2)^(1/2))*x+(1/_C1^2)^(3/2)*(_C1^2*((-18*a*_C1^2*x*y(x)+y(x)^2)/x^2/_C1^2)^(1/2)*x+_C2*x+1/3))/x = 0, 1/(1/_C1^2)^(3/2)*(9*a*ln(((y(x)/x-18*a*_C1^2)/_C1^2+9*a)*_C1+((-18*a*_C1^2*x*y(x)+y(x)^2)/x^2/_C1^2)^(1/2))*x+(1/_C1^2)^(3/2)*(_C1^2*((-18*a*_C1^2*x*y(x)+y(x)^2)/x^2/_C1^2)^(1/2)*x-_C2*x-1/3))/x = 0