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 = 0.330592 (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 = 0.528 (sec), leaf count = 45
\[\left [y \left (x \right ) = \frac {4 \left (\textit {\_C1} x +a^{2}\right )^{\frac {5}{2}}}{15 \textit {\_C1}^{2}}+\textit {\_C2} x +\textit {\_C3}, y \left (x \right ) = -\frac {4 \left (\textit {\_C1} x +a^{2}\right )^{\frac {5}{2}}}{15 \textit {\_C1}^{2}}+\textit {\_C2} x +\textit {\_C3}\right ]\] 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))
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]