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 = 0.794866 (sec), leaf count = 161
\[\text {Solve}\left [\int _1^x\left (\int _1^{y(x)}\frac {2 K[3] \left (e^{K[2] K[3]^2} c_1 \left (K[2] K[3]^2-1\right )+1\right )}{\left (4 K[2] K[3]^2+e^{K[2] K[3]^2} c_1-1\right ){}^2}dK[2]-\frac {2 y(x) K[3]^2+e^{K[3]^2 y(x)} c_1-1}{K[3] \left (4 y(x) K[3]^2+e^{K[3]^2 y(x)} c_1-1\right )}\right )dK[3]+c_2=\int _1^{y(x)}-\frac {x^2}{4 K[2] x^2+e^{x^2 K[2]} c_1-1}dK[2],y(x)\right ]\]
Maple ✓
cpu = 0.688 (sec), leaf count = 32
\[\left [y \left (x \right ) = \frac {\RootOf \left (-\ln \left (x \right )+\textit {\_C2} +\int _{}^{\textit {\_Z}}-\frac {1}{{\mathrm e}^{\textit {\_f}} \textit {\_C1} -4 \textit {\_f} +1}d \textit {\_f} \right )}{x^{2}}\right ]\] 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] + Inactive[Integrate][-((-1 + E^(K[3]^2*y[x])*C[1] + 2*K[3]^2*y[x])/(K[3]*(-1 + E^(K[3]^2*y[x])*C[1] + 4*K[3]^2*y[x]))) + Inactive[Integrate][(2*K[3]*(1 + E^(K[2]*K[3]^2)*C[1]*(-1 + K[2]*K[3]^2)))/(-1 + E^(K[2]*K[3]^2)*C[1] + 4*K[2]*K[3]^2)^2, {K[2], 1, y[x]}], {K[3], 1, x}] == Inactive[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))
Maple raw output
[y(x) = RootOf(-ln(x)+_C2+Intat(-1/(exp(_f)*_C1-4*_f+1),_f = _Z))/x^2]