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 = 0.883822 (sec), leaf count = 75
\[\text {Solve}\left [\frac {\left (\sqrt {c_1} y(x) \sqrt {c_1-\frac {2 a}{y(x)}}+a \log \left (y(x) \left (\sqrt {c_1} \sqrt {c_1-\frac {2 a}{y(x)}}+c_1\right )-a\right )\right ){}^2}{c_1^3}=\left (c_2+x\right ){}^2,y(x)\right ]\]
Maple ✓
cpu = 0.145 (sec), leaf count = 135
\[ \left \{ -{{\it \_C1}}^{2}\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{{{\it \_C1}}^{2}}}-2\,ay \left ( x \right ) }-{a\ln \left ( \left ( {\frac {y \left ( x \right ) }{{{\it \_C1}}^{2}}}-a \right ) {\it \_C1}+\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{{{\it \_C1}}^{2}}}-2\,ay \left ( x \right ) } \right ) \left ( {{\it \_C1}}^{-2} \right ) ^{-{\frac {3}{2}}}}-x-{\it \_C2}=0,{{\it \_C1}}^{2}\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{{{\it \_C1}}^{2}}}-2\,ay \left ( x \right ) }+{a\ln \left ( \left ( {\frac {y \left ( x \right ) }{{{\it \_C1}}^{2}}}-a \right ) {\it \_C1}+\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{{{\it \_C1}}^{2}}}-2\,ay \left ( x \right ) } \right ) \left ( {{\it \_C1}}^{-2} \right ) ^{-{\frac {3}{2}}}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y[x]^2*y''[x] == a,y[x],x]
Mathematica raw output
Solve[(a*Log[-a + (C[1] + Sqrt[C[1]]*Sqrt[C[1] - (2*a)/y[x]])*y[x]] + Sqrt[C[1]]*Sqrt[C[1] - (2*a)/y[x]]*y[x])^2/C[1]^3 == (x + C[2])^2, y[x]]
Maple raw input
dsolve(y(x)^2*diff(diff(y(x),x),x) = a, y(x),'implicit')
Maple raw output
-_C1^2*(1/_C1^2*y(x)^2-2*a*y(x))^(1/2)-a/(1/_C1^2)^(3/2)*ln((1/_C1^2*y(x)-a)*_C1+(1/_C1^2*y(x)^2-2*a*y(x))^(1/2))-x-_C2 = 0, _C1^2*(1/_C1^2*y(x)^2-2*a*y(x))^(1/2)+a/(1/_C1^2)^(3/2)*ln((1/_C1^2*y(x)-a)*_C1+(1/_C1^2*y(x)^2-2*a*y(x))^(1/2))-x-_C2 = 0