ODE
\[ 2 y(x) y''(x)=a+y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.00851739 (sec), leaf count = 29
\[\left \{\left \{y(x)\to \frac {x^2 \left (a+c_1^2\right )}{4 c_2}+c_1 x+c_2\right \}\right \}\]
Maple ✓
cpu = 0.123 (sec), leaf count = 22
\[ \left \{ y \left ( x \right ) ={\frac { \left ( {{\it \_C1}}^{2}+a \right ) {x}^{2}}{4\,{\it \_C2}}}+{\it \_C1}\,x+{\it \_C2} \right \} \] Mathematica raw input
DSolve[2*y[x]*y''[x] == a + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1] + (x^2*(a + C[1]^2))/(4*C[2]) + C[2]}}
Maple raw input
dsolve(2*y(x)*diff(diff(y(x),x),x) = a+diff(y(x),x)^2, y(x),'implicit')
Maple raw output
y(x) = 1/4*(_C1^2+a)/_C2*x^2+_C1*x+_C2