ODE
\[ y(x) \left (f(x) \left (\text {a1}+\text {b1} x^k-1\right )+\text {a2}+\text {b2} x^k+\text {c2} x^{2 k}+f'(x)+f(x)^2\right )+x y'(x) \left (\text {a1}+\text {b1} x^k+2 f(x)\right )+x^2 y''(x)=0 \] ODE Classification
(ODEtools/info) missing specification of intermediate function
Book solution method
TO DO
Mathematica ✗
cpu = 1.85504 (sec), leaf count = 0 , could not solve
DSolve[y[x]*(a2 + b2*x^k + c2*x^(2*k) + (-1 + a1 + b1*x^k)*f[x] + f[x]^2 + Derivative[1][f][x]) + x*(a1 + b1*x^k + 2*f[x])*Derivative[1][y][x] + x^2*Derivative[2][y][x] == 0, y[x], x]
Maple ✗
cpu = 2.788 (sec), leaf count = 0 , result contains DESol
\[ \left \{ y \left ( x \right ) ={\it DESol} \left ( \left \{ {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}{\it \_Y} \left ( x \right ) +{\frac { \left ( {\it a1}+{\it b1}\,{x}^{k}+2\,f \left ( x \right ) \right ) {\frac {\rm d}{{\rm d}x}}{\it \_Y} \left ( x \right ) }{x}}+{\frac { \left ( {\it a2}+{\it b2}\,{x}^{k}+{\it c2}\,{x}^{2\,k}+ \left ( {\it a1}+{\it b1}\,{x}^{k}-1 \right ) f \left ( x \right ) + \left ( f \left ( x \right ) \right ) ^{2}+{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) {\it \_Y} \left ( x \right ) }{{x}^{2}}} \right \} , \left \{ {\it \_Y} \left ( x \right ) \right \} \right ) \right \} \]
Mathematica raw input
DSolve[y[x]*(a2 + b2*x^k + c2*x^(2*k) + (-1 + a1 + b1*x^k)*f[x] + f[x]^2 + f'[x]) + x*(a1 + b1*x^k + 2*f[x])*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
DSolve[y[x]*(a2 + b2*x^k + c2*x^(2*k) + (-1 + a1 + b1*x^k)*f[x] + f[x]^2 + Derivative[1][f][x]) + x*(a1 + b1*x^k + 2*f[x])*Derivative[1][y][x] + x^2*Derivative[2][y][x] == 0, y[x], x]
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+x*(a1+b1*x^k+2*f(x))*diff(y(x),x)+(a2+b2*x^k+c2*x^(2*k)+(a1+b1*x^k-1)*f(x)+f(x)^2+diff(f(x),x))*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = DESol({diff(diff(_Y(x),x),x)+1/x*(a1+b1*x^k+2*f(x))*diff(_Y(x),x)+(a2+b2*x^k+c2*x^(2*k)+(a1+b1*x^k-1)*f(x)+f(x)^2+diff(f(x),x))/x^2*_Y(x)},{_Y(x)})