4.12.15 $y^{\prime }(x)(x(Ax+By(x))+\text{a0}+\text{a1}x+\text{a2}y(x))=y(x)(Ax+By(x))+\text{b0}+\text{b1}x+\text{b2}y(x)$

ODE
$$y^{\prime }(x)(x(Ax+By(x))+\text{a0}+\text{a1}x+\text{a2}y(x))=y(x)(Ax+By(x))+\text{b0}+\text{b1}x+\text{b2}y(x)$$ 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

$$\{-27{\int }^{3\frac{(\mathit{a}\mathit{0}A+B\mathit{b}\mathit{0}-1/3{\mathit{a}\mathit{1}}^2+1/3\mathit{a}\mathit{1}\mathit{b}\mathit{2}-\mathit{a}\mathit{2}\mathit{b}\mathit{1}-1/3{\mathit{b}\mathit{2}}^2)(((-2/3xB+\mathit{a}\mathit{2}/3)\mathit{a}\mathit{1}+(1/3xB+\mathit{a}\mathit{2}/3)\mathit{b}\mathit{2}+A\mathit{a}\mathit{2}x-B\mathit{a}\mathit{0})y\left(x\right)+1/3x{\mathit{a}\mathit{1}}^2+(1/3A{x}^2-2/3\mathit{b}\mathit{2}x+\mathit{a}\mathit{0}/3)\mathit{a}\mathit{1}+(-2/3A{x}^2-2/3\mathit{a}\mathit{0})\mathit{b}\mathit{2}+(\mathit{b}\mathit{1}x+\mathit{b}\mathit{0})(xB+\mathit{a}\mathit{2}))}{(-2/9{\mathit{a}\mathit{1}}^3+1/3{\mathit{a}\mathit{1}}^2\mathit{b}\mathit{2}+(\mathit{a}\mathit{0}A-2B\mathit{b}\mathit{0}-\mathit{a}\mathit{2}\mathit{b}\mathit{1}+1/3{\mathit{b}\mathit{2}}^2)\mathit{a}\mathit{1}-2/9{\mathit{b}\mathit{2}}^3+(-2\mathit{a}\mathit{0}A+B\mathit{b}\mathit{0}-\mathit{a}\mathit{2}\mathit{b}\mathit{1})\mathit{b}\mathit{2}+3A\mathit{a}\mathit{2}\mathit{b}\mathit{0}+3B\mathit{a}\mathit{0}\mathit{b}\mathit{1})((xB+\mathit{a}\mathit{2})y\left(x\right)+A{x}^2+\mathit{a}\mathit{1}x+\mathit{a}\mathit{0})}}\frac{{(3\mathit{a}\mathit{0}A+3B\mathit{b}\mathit{0}-{\mathit{a}\mathit{1}}^2+\mathit{a}\mathit{1}\mathit{b}\mathit{2}-3\mathit{a}\mathit{2}\mathit{b}\mathit{1}-{\mathit{b}\mathit{2}}^2)}^3}{81{(-2/9{\mathit{a}\mathit{1}}^3+1/3{\mathit{a}\mathit{1}}^2\mathit{b}\mathit{2}+(\mathit{a}\mathit{0}A-2B\mathit{b}\mathit{0}-\mathit{a}\mathit{2}\mathit{b}\mathit{1}+1/3{\mathit{b}\mathit{2}}^2)\mathit{a}\mathit{1}-2/9{\mathit{b}\mathit{2}}^3+(-2\mathit{a}\mathit{0}A+B\mathit{b}\mathit{0}-\mathit{a}\mathit{2}\mathit{b}\mathit{1})\mathit{b}\mathit{2}+3A\mathit{a}\mathit{2}\mathit{b}\mathit{0}+3B\mathit{a}\mathit{0}\mathit{b}\mathit{1})}^2{\mathit{\_}\mathit{a}}^3+729{(\mathit{a}\mathit{0}A+B\mathit{b}\mathit{0}-1/3{\mathit{a}\mathit{1}}^2+1/3\mathit{a}\mathit{1}\mathit{b}\mathit{2}-\mathit{a}\mathit{2}\mathit{b}\mathit{1}-1/3{\mathit{b}\mathit{2}}^2)}^3\mathit{\_}\mathit{a}-729{(\mathit{a}\mathit{0}A+B\mathit{b}\mathit{0}-1/3{\mathit{a}\mathit{1}}^2+1/3\mathit{a}\mathit{1}\mathit{b}\mathit{2}-\mathit{a}\mathit{2}\mathit{b}\mathit{1}-1/3{\mathit{b}\mathit{2}}^2)}^3}d\mathit{\_}\mathit{a}-\int -(3\mathit{a}\mathit{0}A+3B\mathit{b}\mathit{0}-{\mathit{a}\mathit{1}}^2+\mathit{a}\mathit{1}\mathit{b}\mathit{2}-3\mathit{a}\mathit{2}\mathit{b}\mathit{1}-{\mathit{b}\mathit{2}}^2)(xB+\mathit{a}\mathit{2}){((-3B^2\mathit{b}\mathit{1}-3A(\mathit{a}\mathit{1}-\mathit{b}\mathit{2})B+3A^2\mathit{a}\mathit{2})x^3+(-3B^2\mathit{b}\mathit{0}+(-6\mathit{a}\mathit{2}\mathit{b}\mathit{1}-3\mathit{a}\mathit{0}A-3\mathit{a}\mathit{1}(\mathit{a}\mathit{1}-\mathit{b}\mathit{2}))B+3A\mathit{a}\mathit{2}(\mathit{a}\mathit{1}+\mathit{b}\mathit{2}))x^2+((-6\mathit{a}\mathit{2}\mathit{b}\mathit{0}-6(\mathit{a}\mathit{1}-\mathit{b}\mathit{2}/2)\mathit{a}\mathit{0})B+3\mathit{a}\mathit{2}(\mathit{a}\mathit{0}A+\mathit{a}\mathit{1}\mathit{b}\mathit{2}-\mathit{a}\mathit{2}\mathit{b}\mathit{1}))x-3B{\mathit{a}\mathit{0}}^2+3\mathit{a}\mathit{0}\mathit{a}\mathit{2}\mathit{b}\mathit{2}-3{\mathit{a}\mathit{2}}^2\mathit{b}\mathit{0})}^{-1}\mathrm{d}x-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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