ODE
\[ x^2 y''(x)=x^4 y'(x)^2-4 x^2 y(x)^2+6 y(x) \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 157.135 (sec), leaf count = 89
\[\text {Solve}\left [\int _1^x -\frac {c_1 e^{K[3]^2}+2 K[3]^2-1}{K[3] \left (c_1 e^{K[3]^2}+4 K[3]^2-1\right )} \, dK[3]+c_2=\int _1^{y(x)} -\frac {x^2}{c_1 e^{x^2 K[2]}+4 x^2 K[2]-1} \, dK[2],y(x)\right ]\]
Maple ✓
cpu = 0.32 (sec), leaf count = 39
\[ \left \{ \ln \left ( x \right ) -{\it \_C2}+\int ^{{x}^{2}y \left ( x \right ) }\!-{\frac {1}{2} \left ( {{\rm e}^{{\it \_f}}} \left ( {\it \_f}-{\frac {1}{2}} \right ) {{\rm e}^{-{\it \_f}}}-{\frac {{{\rm e}^{{\it \_f}}}{\it \_C1}}{2}}+{\it \_f} \right ) ^{-1}}{d{\it \_f}}=0 \right \} \] Mathematica raw input
DSolve[x^2*y''[x] == 6*y[x] - 4*x^2*y[x]^2 + x^4*y'[x]^2,y[x],x]
Mathematica raw output
Solve[C[2] + Integrate[-((-1 + E^K[3]^2*C[1] + 2*K[3]^2)/(K[3]*(-1 + E^K[3]^2*C[1] + 4*K[3]^2))), {K[3], 1, x}] == Integrate[-(x^2/(-1 + E^(x^2*K[2])*C[1] + 4*x^2*K[2])), {K[2], 1, y[x]}], y[x]]
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x) = x^4*diff(y(x),x)^2+6*y(x)-4*x^2*y(x)^2, y(x),'implicit')
Maple raw output
ln(x)-_C2+Intat(-1/2/(exp(_f)*(_f-1/2)*exp(-_f)-1/2*exp(_f)*_C1+_f),_f = x^2*y(x)) = 0