4.41.7 \(a x+y(x)^2 y''(x)+y(x) y'(x)^2=0\)

ODE
\[ a x+y(x)^2 y''(x)+y(x) y'(x)^2=0 \] ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 23.0492 (sec), leaf count = 0 , could not solve

DSolve[a*x + y[x]*Derivative[1][y][x]^2 + y[x]^2*Derivative[2][y][x] == 0, y[x], x]

Maple ✓
cpu = 3.545 (sec), leaf count = 117

\[\left [\ln \left (x \right )-\frac {\sqrt {3}\, \left (\int _{}^{\frac {y \left (x \right )}{x}}\frac {\textit {\_g}^{2} \left (\left (\frac {a}{\textit {\_g}^{3}}\right )^{\frac {1}{3}} \sqrt {3}+3 \left (\frac {a}{\textit {\_g}^{3}}\right )^{\frac {1}{3}} \tan \left (\RootOf \left (-2 \sqrt {3}\, \textit {\_Z} +\ln \left (\frac {\tan ^{2}\left (\textit {\_Z} \right )+1}{\tan ^{2}\left (\textit {\_Z} \right )+2 \sqrt {3}\, \tan \left (\textit {\_Z} \right )+3}\right )+6 \textit {\_C1} +6 \left (\int \frac {\left (\frac {a}{\textit {\_g}^{3}}\right )^{\frac {2}{3}} \textit {\_g}^{2}}{\textit {\_g}^{3}+a}d \textit {\_g} \right )\right )\right )-2 \sqrt {3}\right )}{\textit {\_g}^{3}+a}d \textit {\_g} \right )}{6}-\textit {\_C2} = 0\right ]\] Mathematica raw input

DSolve[a*x + y[x]*y'[x]^2 + y[x]^2*y''[x] == 0,y[x],x]

Mathematica raw output

DSolve[a*x + y[x]*Derivative[1][y][x]^2 + y[x]^2*Derivative[2][y][x] == 0, y[x],
 x]

Maple raw input

dsolve(y(x)^2*diff(diff(y(x),x),x)+y(x)*diff(y(x),x)^2+a*x = 0, y(x))

Maple raw output

[ln(x)-1/6*3^(1/2)*Intat(_g^2*((a/_g^3)^(1/3)*3^(1/2)+3*(a/_g^3)^(1/3)*tan(RootO
f(-2*3^(1/2)*_Z+ln((tan(_Z)^2+1)/(tan(_Z)^2+2*3^(1/2)*tan(_Z)+3))+6*_C1+6*Int((a
/_g^3)^(2/3)/(_g^3+a)*_g^2,_g)))-2*3^(1/2))/(_g^3+a),_g = y(x)/x)-_C2 = 0]