ODE
\[ a x+x y''(x)^2-2 y'(x) y''(x)=0 \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.351783 (sec), leaf count = 261
\[\left \{\left \{y(x)\to c_2-\frac {\sqrt {a} x \sec \left (c_1-i \log (x)\right ) \left (\left (x^2+3\right ) \sin \left (c_1\right )-i \left (x^2-3\right ) \cos \left (c_1\right )\right )}{6 \sqrt {\sec ^2\left (c_1-i \log (x)\right )}}\right \},\left \{y(x)\to c_2+\frac {\sqrt {a} x \sec \left (c_1-i \log (x)\right ) \left (\left (x^2+3\right ) \sin \left (c_1\right )-i \left (x^2-3\right ) \cos \left (c_1\right )\right )}{6 \sqrt {\sec ^2\left (c_1-i \log (x)\right )}}\right \},\left \{y(x)\to c_2-\frac {\sqrt {a} x \sec \left (c_1+i \log (x)\right ) \left (\left (x^2+3\right ) \sin \left (c_1\right )+i \left (x^2-3\right ) \cos \left (c_1\right )\right )}{6 \sqrt {\sec ^2\left (c_1+i \log (x)\right )}}\right \},\left \{y(x)\to c_2+\frac {\sqrt {a} x \sec \left (c_1+i \log (x)\right ) \left (\left (x^2+3\right ) \sin \left (c_1\right )+i \left (x^2-3\right ) \cos \left (c_1\right )\right )}{6 \sqrt {\sec ^2\left (c_1+i \log (x)\right )}}\right \}\right \}\]
Maple ✓
cpu = 0.194 (sec), leaf count = 44
\[ \left \{ y \left ( x \right ) =-{\frac {{x}^{2}}{2}\sqrt {a}}+{\it \_C1},y \left ( x \right ) ={\frac {{x}^{2}}{2}\sqrt {a}}+{\it \_C1},y \left ( x \right ) ={\it \_C2}\,{x}^{3}+{\frac {ax}{12\,{\it \_C2}}}+{\it \_C1} \right \} \] Mathematica raw input
DSolve[a*x - 2*y'[x]*y''[x] + x*y''[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[2] - (Sqrt[a]*x*Sec[C[1] - I*Log[x]]*((-I)*(-3 + x^2)*Cos[C[1]] + (3 + x^2)*Sin[C[1]]))/(6*Sqrt[Sec[C[1] - I*Log[x]]^2])}, {y[x] -> C[2] + (Sqrt[a]*x*Sec[C[1] - I*Log[x]]*((-I)*(-3 + x^2)*Cos[C[1]] + (3 + x^2)*Sin[C[1]]))/(6*Sqrt[Sec[C[1] - I*Log[x]]^2])}, {y[x] -> C[2] - (Sqrt[a]*x*Sec[C[1] + I*Log[x]]*(I*(-3 + x^2)*Cos[C[1]] + (3 + x^2)*Sin[C[1]]))/(6*Sqrt[Sec[C[1] + I*Log[x]]^2])}, {y[x] -> C[2] + (Sqrt[a]*x*Sec[C[1] + I*Log[x]]*(I*(-3 + x^2)*Cos[C[1]] + (3 + x^2)*Sin[C[1]]))/(6*Sqrt[Sec[C[1] + I*Log[x]]^2])}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)^2-2*diff(y(x),x)*diff(diff(y(x),x),x)+a*x = 0, y(x),'implicit')
Maple raw output
y(x) = 1/2*a^(1/2)*x^2+_C1, y(x) = -1/2*a^(1/2)*x^2+_C1, y(x) = _C2*x^3+1/12*a/_C2*x+_C1