4.12.15 \(y'(x) (x (A x+B y(x))+\text {a0}+\text {a1} x+\text {a2} y(x))=y(x) (A x+B y(x))+\text {b0}+\text {b1} x+\text {b2} y(x)\)

ODE
\[ y'(x) (x (A x+B y(x))+\text {a0}+\text {a1} x+\text {a2} y(x))=y(x) (A x+B y(x))+\text {b0}+\text {b1} x+\text {b2} y(x) \] ODE Classification

[_rational, [_Abel, `2nd type``class B`]]

Book solution method
Homogeneous equation, Jacobi equation

Mathematica
cpu = 227.898 (sec), leaf count = 0 , could not solve

DSolve[(a0 + a1*x + a2*y[x] + x*(A*x + B*y[x]))*Derivative[1][y][x] == b0 + b1*x + b2*y[x] + y[x]*(A*x + B*y[x]), y[x], x]

Maple
cpu = 0.285 (sec), leaf count = 509

\[ \left \{ -27\,\int ^{3\,{\frac { \left ({\it a0}\,A+B{\it b0}-1/3\,{{\it a1}}^{2}+1/3\,{\it a1}\,{\it b2}-{\it a2}\,{\it b1}-1/3\,{{\it b2}}^{2} \right ) \left (\left (\left (-2/3\,xB+{\it a2}/3 \right ) {\it a1}+ \left (1/3\,xB+{\it a2}/3 \right ) {\it b2}+A{\it a2}\,x-B{\it a0} \right ) y \relax (x ) +1/3\,x{{\it a1}}^{2}+ \left (1/3\,A{x}^{2}-2/3\,{\it b2}\,x+{\it a0}/3 \right ) {\it a1}+ \left (-2/3\,A{x}^{2}-2/3\,{\it a0} \right ) {\it b2}+ \left ({\it b1}\,x+{\it b0} \right ) \left (xB+{\it a2} \right ) \right ) }{ \left (-2/9\,{{\it a1}}^{3}+1/3\,{{\it a1}}^{2}{\it b2}+ \left ({\it a0}\,A-2\,B{\it b0}-{\it a2}\,{\it b1}+1/3\,{{\it b2}}^{2} \right ) {\it a1}-2/9\,{{\it b2}}^{3}+ \left (-2\,{\it a0}\,A+B{\it b0}-{\it a2}\,{\it b1} \right ) {\it b2}+3\,A{\it a2}\,{\it b0}+3\,B{\it a0}\,{\it b1} \right ) \left (\left (xB+{\it a2} \right ) y \relax (x ) +A{x}^{2}+{\it a1}\,x+{\it a0} \right ) }}}\!{\frac { \left (3\,{\it a0}\,A+3\,B{\it b0}-{{\it a1}}^{2}+{\it a1}\,{\it b2}-3\,{\it a2}\,{\it b1}-{{\it b2}}^{2} \right ) ^{3}}{81\, \left (-2/9\,{{\it a1}}^{3}+1/3\,{{\it a1}}^{2}{\it b2}+ \left ({\it a0}\,A-2\,B{\it b0}-{\it a2}\,{\it b1}+1/3\,{{\it b2}}^{2} \right ) {\it a1}-2/9\,{{\it b2}}^{3}+ \left (-2\,{\it a0}\,A+B{\it b0}-{\it a2}\,{\it b1} \right ) {\it b2}+3\,A{\it a2}\,{\it b0}+3\,B{\it a0}\,{\it b1} \right ) ^{2}{{\it \_a}}^{3}+729\, \left ({\it a0}\,A+B{\it b0}-1/3\,{{\it a1}}^{2}+1/3\,{\it a1}\,{\it b2}-{\it a2}\,{\it b1}-1/3\,{{\it b2}}^{2} \right ) ^{3}{\it \_a}-729\, \left ({\it a0}\,A+B{\it b0}-1/3\,{{\it a1}}^{2}+1/3\,{\it a1}\,{\it b2}-{\it a2}\,{\it b1}-1/3\,{{\it b2}}^{2} \right ) ^{3}}}{d{\it \_a}}-\int \!-{ \left (3\,{\it a0}\,A+3\,B{\it b0}-{{\it a1}}^{2}+{\it a1}\,{\it b2}-3\,{\it a2}\,{\it b1}-{{\it b2}}^{2} \right ) \left (xB+{\it a2} \right ) \left (\left (-3\,{B}^{2}{\it b1}-3\,A \left ({\it a1}-{\it b2} \right ) B+3\,{A}^{2}{\it a2} \right ) {x}^{3}+ \left (-3\,{B}^{2}{\it b0}+ \left (-6\,{\it a2}\,{\it b1}-3\,{\it a0}\,A-3\,{\it a1}\, \left ({\it a1}-{\it b2} \right ) \right ) B+3\,A{\it a2}\, \left ({\it a1}+{\it b2} \right ) \right ) {x}^{2}+ \left (\left (-6\,{\it a2}\,{\it b0}-6\, \left ({\it a1}-{\it b2}/2 \right ) {\it a0} \right ) B+3\,{\it a2}\, \left ({\it a0}\,A+{\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) \right ) x-3\,B{{\it a0}}^{2}+3\,{\it a0}\,{\it a2}\,{\it b2}-3\,{{\it a2}}^{2}{\it b0} \right ) ^{-1}}\,{\rm d}x-{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(a0 + a1*x + a2*y[x] + x*(A*x + B*y[x]))*y'[x] == b0 + b1*x + b2*y[x] + y[x]*(A*x + B*y[x]),y[x],x]

Mathematica raw output

DSolve[(a0 + a1*x + a2*y[x] + x*(A*x + B*y[x]))*Derivative[1][y][x] == b0 + b1*x
 + b2*y[x] + y[x]*(A*x + B*y[x]), y[x], x]

Maple raw input

dsolve((a0+a1*x+a2*y(x)+x*(A*x+B*y(x)))*diff(y(x),x) = b0+b1*x+b2*y(x)+y(x)*(A*x+B*y(x)), y(x),'implicit')

Maple raw output

-27*Intat(1/(81*(-2/9*a1^3+1/3*a1^2*b2+(a0*A-2*B*b0-a2*b1+1/3*b2^2)*a1-2/9*b2^3+
(-2*A*a0+B*b0-a2*b1)*b2+3*A*a2*b0+3*B*a0*b1)^2*_a^3+729*(a0*A+B*b0-1/3*a1^2+1/3*
a1*b2-a2*b1-1/3*b2^2)^3*_a-729*(a0*A+B*b0-1/3*a1^2+1/3*a1*b2-a2*b1-1/3*b2^2)^3)*
(3*A*a0+3*B*b0-a1^2+a1*b2-3*a2*b1-b2^2)^3,_a = 3*(a0*A+B*b0-1/3*a1^2+1/3*a1*b2-a
2*b1-1/3*b2^2)*(((-2/3*x*B+1/3*a2)*a1+(1/3*x*B+1/3*a2)*b2+A*a2*x-B*a0)*y(x)+1/3*
x*a1^2+(1/3*A*x^2-2/3*b2*x+1/3*a0)*a1+(-2/3*A*x^2-2/3*a0)*b2+(b1*x+b0)*(B*x+a2))
/(-2/9*a1^3+1/3*a1^2*b2+(a0*A-2*B*b0-a2*b1+1/3*b2^2)*a1-2/9*b2^3+(-2*A*a0+B*b0-a
2*b1)*b2+3*A*a2*b0+3*B*a0*b1)/((B*x+a2)*y(x)+A*x^2+a1*x+a0))-Int(-(B*x+a2)*(3*A*
a0+3*B*b0-a1^2+a1*b2-3*a2*b1-b2^2)/((-3*B^2*b1-3*A*(a1-b2)*B+3*A^2*a2)*x^3+(-3*B
^2*b0+(-6*a2*b1-3*a0*A-3*a1*(a1-b2))*B+3*A*a2*(a1+b2))*x^2+((-6*a2*b0-6*(a1-1/2*
b2)*a0)*B+3*a2*(A*a0+a1*b2-a2*b1))*x-3*B*a0^2+3*a0*a2*b2-3*a2^2*b0),x)-_C1 = 0