ODE
\[ y(x) y''(x)=a \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.299856 (sec), leaf count = 111
\[\left \{\left \{y(x)\to \exp \left (-\frac {2 a \text {erf}^{-1}\left (-i \sqrt {\frac {2}{\pi }} \sqrt {a e^{\frac {c_1}{a}} (x+c_2){}^2}\right ){}^2+c_1}{2 a}\right )\right \},\left \{y(x)\to \exp \left (-\frac {2 a \text {erf}^{-1}\left (i \sqrt {\frac {2}{\pi }} \sqrt {a e^{\frac {c_1}{a}} (x+c_2){}^2}\right ){}^2+c_1}{2 a}\right )\right \}\right \}\]
Maple ✓
cpu = 0.676 (sec), leaf count = 55
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {2 a \ln \left (\textit {\_a} \right )-2 \textit {\_C1} a}}d \textit {\_a} -x -\textit {\_C2} = 0, \int _{}^{y \left (x \right )}-\frac {1}{\sqrt {2 a \ln \left (\textit {\_a} \right )-2 \textit {\_C1} a}}d \textit {\_a} -x -\textit {\_C2} = 0\right ]\] Mathematica raw input
DSolve[y[x]*y''[x] == a,y[x],x]
Mathematica raw output
{{y[x] -> E^(-1/2*(C[1] + 2*a*InverseErf[(-I)*Sqrt[2/Pi]*Sqrt[a*E^(C[1]/a)*(x + C[2])^2]]^2)/a)}, {y[x] -> E^(-1/2*(C[1] + 2*a*InverseErf[I*Sqrt[2/Pi]*Sqrt[a*E^(C[1]/a)*(x + C[2])^2]]^2)/a)}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = a, y(x))
Maple raw output
[Intat(1/(2*a*ln(_a)-2*_C1*a)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(2*a*ln(_a)-2*_C1*a)^(1/2),_a = y(x))-x-_C2 = 0]