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

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

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type``class A`]]

Book solution method
Equation linear in the variables, \(y'(x)=f\left (\frac {X_1}{X_2} \right ) \)

Mathematica
cpu = 1.4045 (sec), leaf count = 216

\[\text {Solve}\left [\frac {\text {c1}^2 \log \left (\frac {(\text {a2}+(\text {b2}+\text {c2}) y(x))^2 \left (\frac {(\text {b2}+\text {c2}) (\text {a1}+\text {b1} x+\text {c1} y(x)) (\text {a1} (\text {b2}+\text {c2})-\text {a2} \text {c1}+\text {b1} x (\text {b2}+\text {c2}))}{(\text {a2}+(\text {b2}+\text {c2}) y(x))^2}-\text {b1} \text {b2}-\text {b1} \text {c2}\right )}{(\text {a1} (\text {b2}+\text {c2})-\text {a2} \text {c1}+\text {b1} x (\text {b2}+\text {c2}))^2}\right )-\frac {2 \text {c1}^2 \tan ^{-1}\left (\frac {-2 \text {a1} (\text {b2}+\text {c2})+\text {a2} \text {c1}-2 \text {b1} \text {b2} x-2 \text {b1} \text {c2} x-\text {c1} (\text {b2}+\text {c2}) y(x)}{\text {c1} \sqrt {-\frac {4 \text {b1} (\text {b2}+\text {c2})+\text {c1}^2}{\text {c1}^2}} (\text {a2}+(\text {b2}+\text {c2}) y(x))}\right )}{\sqrt {-\frac {4 \text {b1} (\text {b2}+\text {c2})+\text {c1}^2}{\text {c1}^2}}}+2 \text {c1}^2 \log (\text {a1} (\text {b2}+\text {c2})-\text {a2} \text {c1}+\text {b1} x (\text {b2}+\text {c2}))-2 \text {b1} c_1 (\text {b2}+\text {c2})}{2 \text {b1} (\text {b2}+\text {c2})}=0,y(x)\right ]\]

Maple
cpu = 0.043 (sec), leaf count = 228

\[ \left \{ -{\frac {1}{2}\ln \left (-{\frac {{\it b1}\, \left ({\it b2}+{\it c2} \right ) \left (-{\it b1}\, \left ({\it b2}+{\it c2} \right ) ^{2} \left (y \relax (x ) \right ) ^{2}+ \left (\left ({\it b2}+{\it c2} \right ) \left (x{\it c1}-2\,{\it a2} \right ) {\it b1}+{\it c1}\, \left ({\it a1}\,{\it b2}+{\it a1}\,{\it c2}-{\it a2}\,{\it c1} \right ) \right ) y \relax (x ) +{x}^{2} \left ({\it b2}+{\it c2} \right ) {{\it b1}}^{2}+ \left (2\,{\it a1}\,{\it b2}\,x+2\,{\it a1}\,{\it c2}\,x-{\it a2}\,{\it c1}\,x-{{\it a2}}^{2} \right ) {\it b1}+{\it a1}\, \left ({\it a1}\,{\it b2}+{\it a1}\,{\it c2}-{\it a2}\,{\it c1} \right ) \right ) }{ \left (\left ({\it b2}+{\it c2} \right ) {\it a1}+x \left ({\it b2}+{\it c2} \right ) {\it b1}-{\it a2}\,{\it c1} \right ) ^{2}}} \right ) }-{{\it c1}{\it Artanh} \left ({\frac {-2\,{\it b1}\, \left ({\it b2}+{\it c2} \right ) ^{2}y \relax (x ) + \left ({\it b2}+{\it c2} \right ) \left (x{\it c1}-2\,{\it a2} \right ) {\it b1}+{\it c1}\, \left ({\it a1}\,{\it b2}+{\it a1}\,{\it c2}-{\it a2}\,{\it c1} \right ) }{ \left ({\it b2}+{\it c2} \right ) {\it a1}+x \left ({\it b2}+{\it c2} \right ) {\it b1}-{\it a2}\,{\it c1}}{\frac {1}{\sqrt {4\,{\it b1}\,{\it b2}+4\,{\it b1}\,{\it c2}+{{\it c1}}^{2}}}}} \right ) {\frac {1}{\sqrt {4\,{\it b1}\,{\it b2}+4\,{\it b1}\,{\it c2}+{{\it c1}}^{2}}}}}-\ln \left (\left ({\it b2}+{\it c2} \right ) {\it a1}+x \left ({\it b2}+{\it c2} \right ) {\it b1}-{\it a2}\,{\it c1} \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

Solve[((-2*c1^2*ArcTan[(a2*c1 - 2*a1*(b2 + c2) - 2*b1*b2*x - 2*b1*c2*x - c1*(b2 
+ c2)*y[x])/(c1*Sqrt[-((c1^2 + 4*b1*(b2 + c2))/c1^2)]*(a2 + (b2 + c2)*y[x]))])/S
qrt[-((c1^2 + 4*b1*(b2 + c2))/c1^2)] - 2*b1*(b2 + c2)*C[1] + 2*c1^2*Log[-(a2*c1)
 + a1*(b2 + c2) + b1*(b2 + c2)*x] + c1^2*Log[((a2 + (b2 + c2)*y[x])^2*(-(b1*b2) 
- b1*c2 + ((b2 + c2)*(-(a2*c1) + a1*(b2 + c2) + b1*(b2 + c2)*x)*(a1 + b1*x + c1*
y[x]))/(a2 + (b2 + c2)*y[x])^2))/(-(a2*c1) + a1*(b2 + c2) + b1*(b2 + c2)*x)^2])/
(2*b1*(b2 + c2)) == 0, y[x]]

Maple raw input

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

Maple raw output

-1/2*ln(-(b2+c2)*b1*(-b1*(b2+c2)^2*y(x)^2+((b2+c2)*(c1*x-2*a2)*b1+c1*(a1*b2+a1*c
2-a2*c1))*y(x)+x^2*(b2+c2)*b1^2+(2*a1*b2*x+2*a1*c2*x-a2*c1*x-a2^2)*b1+a1*(a1*b2+
a1*c2-a2*c1))/((b2+c2)*a1+x*(b2+c2)*b1-a2*c1)^2)-1/(4*b1*b2+4*b1*c2+c1^2)^(1/2)*
arctanh((-2*b1*(b2+c2)^2*y(x)+(b2+c2)*(c1*x-2*a2)*b1+c1*(a1*b2+a1*c2-a2*c1))/(4*
b1*b2+4*b1*c2+c1^2)^(1/2)/((b2+c2)*a1+x*(b2+c2)*b1-a2*c1))*c1-ln((b2+c2)*a1+x*(b
2+c2)*b1-a2*c1)-_C1 = 0