ODE
\[ 2 x y''(x) y'''(x)=y''(x)^2-a^2 \] ODE Classification
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0643706 (sec), leaf count = 75
\[\left \{\left \{y(x)\to -\frac {4}{15} e^{-4 c_1} \left (a^2+e^{2 c_1} x\right ){}^{5/2}+c_3 x+c_2\right \},\left \{y(x)\to \frac {4}{15} e^{-4 c_1} \left (a^2+e^{2 c_1} x\right ){}^{5/2}+c_3 x+c_2\right \}\right \}\]
Maple ✓
cpu = 0.119 (sec), leaf count = 45
\[ \left \{ y \left ( x \right ) =-{\frac {4}{15\,{{\it \_C1}}^{2}} \left ( {\it \_C1}\,x+{a}^{2} \right ) ^{{\frac {5}{2}}}}+{\it \_C2}\,x+{\it \_C3},y \left ( x \right ) ={\frac {4}{15\,{{\it \_C1}}^{2}} \left ( {\it \_C1}\,x+{a}^{2} \right ) ^{{\frac {5}{2}}}}+{\it \_C2}\,x+{\it \_C3} \right \} \] Mathematica raw input
DSolve[2*x*y''[x]*y'''[x] == -a^2 + y''[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> (-4*(a^2 + E^(2*C[1])*x)^(5/2))/(15*E^(4*C[1])) + C[2] + x*C[3]}, {y[x] -> (4*(a^2 + E^(2*C[1])*x)^(5/2))/(15*E^(4*C[1])) + C[2] + x*C[3]}}
Maple raw input
dsolve(2*x*diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x) = diff(diff(y(x),x),x)^2-a^2, y(x),'implicit')
Maple raw output
y(x) = 4/15*(_C1*x+a^2)^(5/2)/_C1^2+_C2*x+_C3, y(x) = -4/15*(_C1*x+a^2)^(5/2)/_C1^2+_C2*x+_C3