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 = 0.0540693 (sec), leaf count = 116
\[\left \{\left \{y(x)\to -\frac {1}{2} x \sqrt {a^2 x^2+2 c_1}-\frac {c_1 \log \left (a \left (\sqrt {a^2 x^2+2 c_1}+a x\right )\right )}{a}+c_2\right \},\left \{y(x)\to \frac {1}{2} x \sqrt {a^2 x^2+2 c_1}+\frac {c_1 \log \left (a \left (\sqrt {a^2 x^2+2 c_1}+a x\right )\right )}{a}+c_2\right \}\right \}\]
Maple ✓
cpu = 0.167 (sec), leaf count = 101
\[ \left \{ y \left ( x \right ) =-{\frac {x}{2}\sqrt {{a}^{2}{x}^{2}+{\it \_C1}}}-{\frac {{\it \_C1}}{2}\ln \left ( {{a}^{2}x{\frac {1}{\sqrt {{a}^{2}}}}}+\sqrt {{a}^{2}{x}^{2}+{\it \_C1}} \right ) {\frac {1}{\sqrt {{a}^{2}}}}}+{\it \_C2},y \left ( x \right ) ={\frac {x}{2}\sqrt {{a}^{2}{x}^{2}+{\it \_C1}}}+{\frac {{\it \_C1}}{2}\ln \left ( {{a}^{2}x{\frac {1}{\sqrt {{a}^{2}}}}}+\sqrt {{a}^{2}{x}^{2}+{\it \_C1}} \right ) {\frac {1}{\sqrt {{a}^{2}}}}}+{\it \_C2} \right \} \] Mathematica raw input
DSolve[y'[x]*y''[x] == a^2*x,y[x],x]
Mathematica raw output
{{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}, {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),'implicit')
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