4.12.16 \(y'(x) (x (\text {a1}+\text {b2} x+\text {c2} y(x))+\text {a1}+\text {b1} x+\text {c1} y(x))=y(x) (\text {a2}+\text {b2} x+\text {c2} y(x))+\text {a3}+\text {b3} x+\text {c2} y(x)\)

ODE
\[ y'(x) (x (\text {a1}+\text {b2} x+\text {c2} y(x))+\text {a1}+\text {b1} x+\text {c1} y(x))=y(x) (\text {a2}+\text {b2} x+\text {c2} y(x))+\text {a3}+\text {b3} x+\text {c2} y(x) \] ODE Classification

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

Book solution method
Homogeneous equation, Jacobi equation

Mathematica
cpu = 273.829 (sec), leaf count = 0 , could not solve

DSolve[(a1 + b1*x + c1*y[x] + x*(a1 + b2*x + c2*y[x]))*Derivative[1][y][x] == a3 + b3*x + c2*y[x] + y[x]*(a2 + b2*x + c2*y[x]), y[x], x]

Maple
cpu = 0.847 (sec), leaf count = 3737 \[ \text {Too large to display} \] Mathematica raw input

DSolve[(a1 + b1*x + c1*y[x] + x*(a1 + b2*x + c2*y[x]))*y'[x] == a3 + b3*x + c2*y[x] + y[x]*(a2 + b2*x + c2*y[x]),y[x],x]

Mathematica raw output

DSolve[(a1 + b1*x + c1*y[x] + x*(a1 + b2*x + c2*y[x]))*Derivative[1][y][x] == a3
 + b3*x + c2*y[x] + y[x]*(a2 + b2*x + c2*y[x]), y[x], x]

Maple raw input

dsolve((a1+b1*x+c1*y(x)+x*(a1+b2*x+c2*y(x)))*diff(y(x),x) = a3+b3*x+c2*y(x)+y(x)*(a2+b2*x+c2*y(x)), y(x),'implicit')

Maple raw output

27*Intat((a1^2+(-a2+2*b1-3*b2-c2)*a1+c2^2+(2*a2-3*a3-b1)*c2+a2^2-a2*b1+b1^2+3*b3
*c1)^3/((4*_a^3-27*_a+27)*a1^6-12*(3+_a)*(a2-2*b1+3*b2+c2)*(_a-3/2)^2*a1^5+((-3*
_a^3-162*_a+162)*c2^2+((-6*a2+72*a3-60*b1+126*b2-108*b3)*_a^3+(-324*a2+243*a3+40
5*b1-486*b2)*_a+324*a2-243*a3-405*b1+486*b2)*c2+(-3*a2^2+(-60*b1+126*b2)*a2+60*b
1^2-144*b1*b2+81*b2^2+36*b3*c1)*_a^3+(-162*a2^2+(405*b1-486*b2)*a2-405*b1^2+972*
b1*b2-729*b2^2-243*b3*c1)*_a+162*a2^2+(-405*b1+486*b2)*a2+405*b1^2-972*b1*b2+729
*b2^2+243*b3*c1)*a1^4+((26*_a^3+189*_a-189)*c2^3+((78*a2-144*a3-12*b1-54*b2+162*
b3)*_a^3+(567*a2-486*a3-648*b1+729*b2)*_a-567*a2+486*a3+648*b1-729*b2)*c2^2+((78
*a2^2+(-144*a3-24*b1-108*b2+162*b3)*a2-120*b1^2+(288*a3+378*b2-324*b3)*b1-324*a3
*b2-324*b2^2+486*b2*b3-18*b3*c1)*_a^3+(567*a2^2+(-486*a3-1296*b1+1458*b2)*a2+810
*b1^2+(972*a3-1458*b2)*b1-1458*a3*b2+729*b2^2+486*b3*c1)*_a-567*a2^2+(486*a3+129
6*b1-1458*b2)*a2-810*b1^2+(-972*a3+1458*b2)*b1+1458*a3*b2-729*b2^2-486*b3*c1)*c2
+(26*a2^3+(-12*b1-54*b2)*a2^2+(-120*b1^2+378*b1*b2-324*b2^2-18*b3*c1)*a2+80*b1^3
-216*b2*b1^2+(162*b2^2+144*b3*c1)*b1-108*b2*c1*(a3+3/2*b3))*_a^3+189*(a2^2+(-10/
7*b1+6/7*b2)*a2+10/7*b1^2-12/7*b1*b2+9/7*b2^2+18/7*b3*c1)*(a2-2*b1+3*b2)*_a-189*
(a2^2+(-10/7*b1+6/7*b2)*a2+10/7*b1^2-12/7*b1*b2+9/7*b2^2+18/7*b3*c1)*(a2-2*b1+3*
b2))*a1^3+((-3*_a^3-162*_a+162)*c2^4+((-12*a2-54*a3+78*b1-144*b2+162*b3)*_a^3+(-
648*a2+729*a3+567*b1-486*b2)*_a+648*a2-729*a3-567*b1+486*b2)*c2^3+((-18*a2^2+(-1
08*a3+234*b1-432*b2+324*b3)*a2-18*b1^2+(-432*a3-108*b2+324*b3)*b1+324*a3^2+(810*
b2-972*b3)*a3+324*b2^2-972*b2*b3+729*b3*(b3-4/27*c1))*_a^3+(-972*a2^2+(1458*a3+1
701*b1-1458*b2)*a2-972*b1^2+(-1458*a3+1458*b2)*b1-729*a3^2+1458*a3*b2-729*b2^2-7
29*b3*c1)*_a+972*a2^2+(-1458*a3-1701*b1+1458*b2)*a2+972*b1^2+(1458*a3-1458*b2)*b
1+729*a3^2-1458*a3*b2+729*b2^2+729*b3*c1)*c2^2+((-12*a2^3+(-54*a3+234*b1-432*b2+
162*b3)*a2^2+(-36*b1^2+(-432*a3-216*b2+324*b3)*b1+810*a3*b2+648*b2^2-972*b2*b3-2
16*b3*c1)*a2-120*b1^3+(432*a3+378*b2-324*b3)*b1^2+(-648*a3*b2-324*b2^2+486*b2*b3
-54*b3*c1)*b1+162*c1*((b2+2*b3)*a3+b3*(b2-3*b3)))*_a^3+(-648*a2^3+(729*a3+1701*b
1-1458*b2)*a2^2+(-1944*b1^2+(-1458*a3+2916*b2)*b1+1458*a3*b2-1458*b2^2-1458*b3*c
1)*a2+810*b1^3+(1458*a3-1458*b2)*b1^2+(-2916*a3*b2+729*b2^2+1458*b3*c1)*b1+(2187
*b2^2+1458*b3*c1)*a3-1458*b2*b3*c1)*_a+648*a2^3+(-729*a3-1701*b1+1458*b2)*a2^2+(
1944*b1^2+(1458*a3-2916*b2)*b1-1458*a3*b2+1458*b2^2+1458*b3*c1)*a2-810*b1^3+(-14
58*a3+1458*b2)*b1^2+(2916*a3*b2-729*b2^2-1458*b3*c1)*b1+(-2187*b2^2-1458*b3*c1)*
a3+1458*b2*b3*c1)*c2+(-3*a2^4+(78*b1-144*b2)*a2^3+(-18*b1^2-108*b1*b2+324*b2^2-1
08*b3*c1)*a2^2+(-120*b1^3+378*b2*b1^2+(-324*b2^2-54*b3*c1)*b1+162*b2*c1*(a3+b3))
*a2+60*b1^4-144*b2*b1^3+(81*b2^2+216*b3*c1)*b1^2-324*b2*c1*(a3+b3)*b1+486*a3*b2^
2*c1+81*b3^2*c1^2)*_a^3-162*(a2^2+(-5/2*b1+3*b2)*a2+5/2*b1^2-6*b1*b2+9/2*b2^2+3/
2*b3*c1)*(a2^2-a2*b1+b1^2+3*b3*c1)*_a+162*(a2^2+(-5/2*b1+3*b2)*a2+5/2*b1^2-6*b1*
b2+9/2*b2^2+3/2*b3*c1)*(a2^2-a2*b1+b1^2+3*b3*c1))*a1^2+((-12*_a^3+81*_a-81)*c2^5
+((-60*a2+126*a3-6*b1+72*b2-108*b3)*_a^3+(405*a2-486*a3-324*b1+243*b2)*_a-405*a2
+486*a3+324*b1-243*b2)*c2^4+((-120*a2^2+(378*a3-24*b1+288*b2-324*b3)*a2+78*b1^2+
(-108*a3-144*b2+162*b3)*b1-324*a3^2+(-324*b2+486*b3)*a3-18*b3*c1)*_a^3+(810*a2^2
+(-1458*a3-1296*b1+972*b2)*a2+567*b1^2+(1458*a3-486*b2)*b1+729*a3^2-1458*a3*b2+4
86*b3*c1)*_a-810*a2^2+(1458*a3+1296*b1-972*b2)*a2-567*b1^2+(-1458*a3+486*b2)*b1-
729*a3^2+1458*a3*b2-486*b3*c1)*c2^3+((-120*a2^3+(378*a3-36*b1+432*b2-324*b3)*a2^
2+(234*b1^2+(-216*a3-432*b2+324*b3)*b1-324*a3^2+(-648*b2+486*b3)*a3-54*b3*c1)*a2
-12*b1^3+(-432*a3-54*b2+162*b3)*b1^2+(648*a3^2+(810*b2-972*b3)*a3-216*b3*c1)*b1+
162*c1*((b2+b3)*a3+2*b2*b3-3*b3^2))*_a^3+(810*a2^3+(-1458*a3-1944*b1+1458*b2)*a2
^2+(1701*b1^2+(2916*a3-1458*b2)*b1+729*a3^2-2916*a3*b2+1458*b3*c1)*a2-648*b1^3+(
-1458*a3+729*b2)*b1^2+(-1458*a3^2+1458*a3*b2-1458*b3*c1)*b1+2187*a3^2*b2-1458*a3
*b3*c1+1458*b2*b3*c1)*_a-810*a2^3+(1458*a3+1944*b1-1458*b2)*a2^2+(-1701*b1^2+(-2
916*a3+1458*b2)*b1-729*a3^2+2916*a3*b2-1458*b3*c1)*a2+648*b1^3+(1458*a3-729*b2)*
b1^2+(1458*a3^2-1458*a3*b2+1458*b3*c1)*b1-2187*a3^2*b2+1458*a3*b3*c1-1458*b2*b3*
c1)*c2^2+((-60*a2^4+(126*a3-24*b1+288*b2-108*b3)*a2^3+(234*b1^2+(-108*a3-432*b2+
162*b3)*b1-324*a3*b2-54*b3*c1)*a2^2+(-24*b1^3+(-432*a3-108*b2+162*b3)*b1^2+(810*
a3*b2-432*b3*c1)*b1+324*((b2+1/2*b3)*a3+2*(b2-3/4*b3)*b3)*c1)*a2-60*b1^4+(288*a3
+126*b2-108*b3)*b1^3+(-324*a3*b2-54*b3*c1)*b1^2+324*((b2+2*b3)*a3+1/2*b3*(b2-3*b
3))*c1*b1-972*c1*(a3^2*b2+b2*(b2-3/2*b3)*a3-1/6*b3^2*c1))*_a^3+405*(a2^2-a2*b1+b
1^2+3*b3*c1)*(a2^2+(-6/5*a3-11/5*b1+12/5*b2)*a2+b1^2+(12/5*a3-6/5*b2)*b1-18/5*a3
*b2+3/5*b3*c1)*_a-405*(a2^2-a2*b1+b1^2+3*b3*c1)*(a2^2+(-6/5*a3-11/5*b1+12/5*b2)*
a2+b1^2+(12/5*a3-6/5*b2)*b1-18/5*a3*b2+3/5*b3*c1))*c2-12*(a2^2+(2*b1-6*b2)*a2-2*
b1^2+3*b1*b2-3*b3*c1)*(a2^3-3/2*a2^2*b1+(-3/2*b1^2+9/2*b3*c1)*a2-27/2*a3*c1*b2+b
1^3+9/2*b1*b3*c1)*_a^3+81*(a2^2-a2*b1+b1^2+3*b3*c1)^2*(a2-2*b1+3*b2)*_a-81*(a2^2
-a2*b1+b1^2+3*b3*c1)^2*(a2-2*b1+3*b2))*a1+(4*_a^3-27*_a+27)*c2^6+24*(a2-3/2*a3-1
/2*b1)*(3+_a)*(_a-3/2)^2*c2^5+((60*a2^2+(-144*a3-60*b1)*a2+81*a3^2+126*a3*b1-3*b
1^2+36*b3*c1)*_a^3+(-405*a2^2+(972*a3+405*b1)*a2-729*a3^2-486*a3*b1-162*b1^2-243
*b3*c1)*_a+405*a2^2+(-972*a3-405*b1)*a2+729*a3^2+486*a3*b1+162*b1^2+243*b3*c1)*c
2^4+((80*a2^3+(-216*a3-120*b1)*a2^2+(162*a3^2+378*a3*b1-12*b1^2+144*b3*c1)*a2+26
*b1^3-54*a3*b1^2+(-324*a3^2-18*b3*c1)*b1-108*a3*c1*(b2+3/2*b3))*_a^3-540*(a2^2+(
-6/5*a3-b1)*a2+9/10*a3^2+3/5*a3*b1+7/10*b1^2+9/5*b3*c1)*(a2-3/2*a3-1/2*b1)*_a+54
0*(a2^2+(-6/5*a3-b1)*a2+9/10*a3^2+3/5*a3*b1+7/10*b1^2+9/5*b3*c1)*(a2-3/2*a3-1/2*
b1))*c2^3+((60*a2^4+(-144*a3-120*b1)*a2^3+(81*a3^2+378*a3*b1-18*b1^2+216*b3*c1)*
a2^2+(78*b1^3-108*a3*b1^2+(-324*a3^2-54*b3*c1)*b1-324*a3*c1*(b2+b3))*a2-3*b1^4-1
44*a3*b1^3+(324*a3^2-108*b3*c1)*b1^2+162*a3*c1*(b2+b3)*b1+486*a3^2*b2*c1+81*b3^2
*c1^2)*_a^3-405*(a2^2+(-12/5*a3-b1)*a2+9/5*a3^2+6/5*a3*b1+2/5*b1^2+3/5*b3*c1)*(a
2^2-a2*b1+b1^2+3*b3*c1)*_a+405*(a2^2+(-12/5*a3-b1)*a2+9/5*a3^2+6/5*a3*b1+2/5*b1^
2+3/5*b3*c1)*(a2^2-a2*b1+b1^2+3*b3*c1))*c2^2+(24*(a2^2+(-3/2*a3-b1)*a2+3*a3*b1-1
/2*b1^2+3/2*b3*c1)*(a2^3-3/2*a2^2*b1+(-3/2*b1^2+9/2*b3*c1)*a2-27/2*a3*c1*b2+b1^3
+9/2*b1*b3*c1)*_a^3-162*(a2-3/2*a3-1/2*b1)*(a2^2-a2*b1+b1^2+3*b3*c1)^2*_a+162*(a
2-3/2*a3-1/2*b1)*(a2^2-a2*b1+b1^2+3*b3*c1)^2)*c2+4*(a2^3-3/2*a2^2*b1+(-3/2*b1^2+
9/2*b3*c1)*a2-27/2*a3*c1*b2+b1^3+9/2*b1*b3*c1)^2*_a^3-27*(a2^2-a2*b1+b1^2+3*b3*c
1)^3*_a+27*(a2^2-a2*b1+b1^2+3*b3*c1)^3),_a = 3/2*((c2^2*x+((-2*x-3)*a1+(a2-2*b1)
*x+c1)*c2+3*(b2*x+1/3*a1+1/3*a2+1/3*b1)*c1)*y(x)+((-2*x-2)*a1-2*x*((b2-3/2*b3)*x
-3/2*a3+b1))*c2+(1+x)*a1^2+(b2*x^2+(-2*a2+2*b1)*x-2*a2+b1)*a1-2*b2*(a2-1/2*b1)*x
^2+(-2*a2*b1+b1^2+3*b3*c1)*x+3*a3*c1)*(c2^2+(-a1+2*a2-3*a3-b1)*c2+3*b3*c1+a1^2+(
-a2+2*b1-3*b2)*a1+a2^2-a2*b1+b1^2)/((c2*x+c1)*y(x)+b2*x^2+a1*x+b1*x+a1)/(c2^3+(-
3/2*a1-9/2*a3-3/2*b1+3*a2)*c2^2+(-3/2*a1^2+(-3*b1+9*b2-3*a2-27/2*b3+9*a3)*a1-3/2
*b1^2+(-3*a2+9*a3)*b1+3*a2^2-9/2*a2*a3+9/2*b3*c1)*c2+a1^3+(-3/2*a2-9/2*b2+3*b1)*
a1^2+(3*b1^2+(-3*a2-9/2*b2)*b1-3/2*a2^2+9*b2*a2+9/2*b3*c1)*a1+b1^3-3/2*a2*b1^2+(
-3/2*a2^2+9/2*b3*c1)*b1+a2^3+9/2*a2*b3*c1-27/2*a3*c1*b2))-Int(-(c2*x+c1)*(c2^2+(
-a1+2*a2-3*a3-b1)*c2+3*b3*c1+a1^2+(-a2+2*b1-3*b2)*a1+a2^2-a2*b1+b1^2)/(((-3*b2+3
*b3)*c2^2+3*b2*(a1-a2+b1)*c2-3*b2^2*c1)*x^3+((-3*a1+3*a3-3*b1)*c2^2+((-3*b2+6*b3
)*c1+3*a1^2+(-3*a2+6*b1+3*b2)*a1-3*b1*(a2-b1))*c2-3*b2*c1*(a1+a2+b1))*x^2+(-3*a1
*c2^2+((-3*a1+6*a3-3*b1)*c1+6*(a1-1/2*a2+b1)*a1)*c2-3*c1*(-b3*c1+(a2+b2)*a1+a2*b
1))*x+3*a1*(a1-c1)*c2-3*a1*a2*c1+3*a3*c1^2),x)-_C1 = 0