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 = 0.209895 (sec), leaf count = 111
\[\left \{\left \{y(x)\to \exp \left (-\frac {c_1+2 a \text {erf}^{-1}\left (-i \sqrt {\frac {2}{\pi }} \sqrt {a e^{\frac {c_1}{a}} \left (c_2+x\right ){}^2}\right ){}^2}{2 a}\right )\right \},\left \{y(x)\to \exp \left (-\frac {c_1+2 a \text {erf}^{-1}\left (i \sqrt {\frac {2}{\pi }} \sqrt {a e^{\frac {c_1}{a}} \left (c_2+x\right ){}^2}\right ){}^2}{2 a}\right )\right \}\right \}\]
Maple ✓
cpu = 0.389 (sec), leaf count = 54
\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {2\,a\ln \left ( {\it \_a} \right ) -2\,{\it \_C1}\,a}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{\sqrt {-2\,a \left ( {\it \_C1}-\ln \left ( {\it \_a} \right ) \right ) }}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y[x]*y''[x] == a,y[x],x]
Mathematica raw output
{{y[x] -> E^(-(C[1] + 2*a*InverseErf[(-I)*Sqrt[2/Pi]*Sqrt[a*E^(C[1]/a)*(x + C[2])^2]]^2)/(2*a))}, {y[x] -> E^(-(C[1] + 2*a*InverseErf[I*Sqrt[2/Pi]*Sqrt[a*E^(C[1]/a)*(x + C[2])^2]]^2)/(2*a))}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x) = a, y(x),'implicit')
Maple raw output
Intat(1/(2*a*ln(_a)-2*_C1*a)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(-2*a*(_C1-ln(_a)))^(1/2),_a = y(x))-x-_C2 = 0