ODE
\[ y'(x) y''(x)=a^2 x \] ODE Classification
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.205358 (sec), leaf count = 116
\[\left \{\left \{y(x)\to -\frac {1}{2} x \sqrt {a^2 x^2+2 c_1}-\frac {c_1 \log \left (a \left (a x+\sqrt {a^2 x^2+2 c_1}\right )\right )}{a}+c_2\right \},\left \{y(x)\to \frac {1}{2} x \sqrt {a^2 x^2+2 c_1}+\frac {c_1 \log \left (a \left (a x+\sqrt {a^2 x^2+2 c_1}\right )\right )}{a}+c_2\right \}\right \}\]
Maple ✓
cpu = 0.577 (sec), leaf count = 101
\[\left [y \left (x \right ) = \frac {x \sqrt {a^{2} x^{2}+\textit {\_C1}}}{2}+\frac {\textit {\_C1} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+\textit {\_C1}}\right )}{2 \sqrt {a^{2}}}+\textit {\_C2}, y \left (x \right ) = -\frac {x \sqrt {a^{2} x^{2}+\textit {\_C1}}}{2}-\frac {\textit {\_C1} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+\textit {\_C1}}\right )}{2 \sqrt {a^{2}}}+\textit {\_C2}\right ]\] Mathematica raw input
DSolve[y'[x]*y''[x] == a^2*x,y[x],x]
Mathematica raw output
{{y[x] -> -1/2*(x*Sqrt[a^2*x^2 + 2*C[1]]) + C[2] - (C[1]*Log[a*(a*x + Sqrt[a^2*x^2 + 2*C[1]])])/a}, {y[x] -> (x*Sqrt[a^2*x^2 + 2*C[1]])/2 + C[2] + (C[1]*Log[a*(a*x + Sqrt[a^2*x^2 + 2*C[1]])])/a}}
Maple raw input
dsolve(diff(y(x),x)*diff(diff(y(x),x),x) = a^2*x, y(x))
Maple raw output
[y(x) = 1/2*x*(a^2*x^2+_C1)^(1/2)+1/2*_C1*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2+_C1)^(1/2))/(a^2)^(1/2)+_C2, y(x) = -1/2*x*(a^2*x^2+_C1)^(1/2)-1/2*_C1*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2+_C1)^(1/2))/(a^2)^(1/2)+_C2]