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 = 0.471435 (sec), leaf count = 261
\[\left \{\left \{y(x)\to c_2-\frac {\sqrt {a} x \sec (c_1-i \log (x)) \left (\left (x^2+3\right ) \sin (c_1)-i \left (x^2-3\right ) \cos (c_1)\right )}{6 \sqrt {\sec ^2(c_1-i \log (x))}}\right \},\left \{y(x)\to \frac {\sqrt {a} x \sec (c_1-i \log (x)) \left (\left (x^2+3\right ) \sin (c_1)-i \left (x^2-3\right ) \cos (c_1)\right )}{6 \sqrt {\sec ^2(c_1-i \log (x))}}+c_2\right \},\left \{y(x)\to c_2-\frac {\sqrt {a} x \sec (i \log (x)+c_1) \left (i \left (x^2-3\right ) \cos (c_1)+\left (x^2+3\right ) \sin (c_1)\right )}{6 \sqrt {\sec ^2(i \log (x)+c_1)}}\right \},\left \{y(x)\to \frac {\sqrt {a} x \sec (i \log (x)+c_1) \left (i \left (x^2-3\right ) \cos (c_1)+\left (x^2+3\right ) \sin (c_1)\right )}{6 \sqrt {\sec ^2(i \log (x)+c_1)}}+c_2\right \}\right \}\]
Maple ✓
cpu = 2.652 (sec), leaf count = 44
\[\left [y \left (x \right ) = \frac {\sqrt {a}\, x^{2}}{2}+\textit {\_C1}, y \left (x \right ) = -\frac {\sqrt {a}\, x^{2}}{2}+\textit {\_C1}, y \left (x \right ) = \textit {\_C2} \,x^{3}+\frac {a x}{12 \textit {\_C2}}+\textit {\_C1}\right ]\] 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))
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]