4.11.6 \(\text {a1}+y'(x) (\text {a2}+b x+\text {c2} y(x))+b y(x)+\text {b1} x=0\)

ODE
\[ \text {a1}+y'(x) (\text {a2}+b x+\text {c2} y(x))+b y(x)+\text {b1} x=0 \] ODE Classification

[[_homogeneous, `class C`], _exact, _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 = 0.0450018 (sec), leaf count = 106

\[\left \{\left \{y(x)\to -\frac {\frac {\sqrt {-x (2 \text {a1}+\text {b1} x)+\frac {(\text {a2}+b x)^2}{\text {c2}}+c_1 \text {c2}}}{\sqrt {\frac {1}{\text {c2}}}}+\text {a2}+b x}{\text {c2}}\right \},\left \{y(x)\to \sqrt {\frac {1}{\text {c2}}} \sqrt {-x (2 \text {a1}+\text {b1} x)+\frac {(\text {a2}+b x)^2}{\text {c2}}+c_1 \text {c2}}-\frac {\text {a2}+b x}{\text {c2}}\right \}\right \}\]

Maple ✓
cpu = 0.037 (sec), leaf count = 126

\[ \left \{ -{\frac {1}{2}\ln \left ( 2\,{\frac { \left ( {b}^{2}-{\it b1}\,{\it c2} \right ) \left ( 1/2\,{\it c2}\, \left ( {b}^{2}-{\it b1}\,{\it c2} \right ) \left ( y \left ( x \right ) \right ) ^{2}+ \left ( {b}^{2}-{\it b1}\,{\it c2} \right ) \left ( bx+{\it a2} \right ) y \left ( x \right ) + \left ( 1/2\,{x}^{2}{\it b1}+{\it a1}\,x \right ) {b}^{2}+{\it a1}\,{\it a2}\,b-1/2\, \left ( {\it b1}\,x+{\it a1} \right ) ^{2}{\it c2}-1/2\,{{\it a2}}^{2}{\it b1} \right ) }{ \left ( {b}^{2}x+{\it a2}\,b- \left ( {\it b1}\,x+{\it a1} \right ) {\it c2} \right ) ^{2}}} \right ) }-\ln \left ( \left ( {\it b1}\,x+{\it a1} \right ) {\it c2}-{b}^{2}x-{\it a2}\,b \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[a1 + b1*x + b*y[x] + (a2 + b*x + c2*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((a2 + b*x + Sqrt[(a2 + b*x)^2/c2 - x*(2*a1 + b1*x) + c2*C[1]]/Sqrt[c
2^(-1)])/c2)}, {y[x] -> -((a2 + b*x)/c2) + Sqrt[c2^(-1)]*Sqrt[(a2 + b*x)^2/c2 - 
x*(2*a1 + b1*x) + c2*C[1]]}}

Maple raw input

dsolve((a2+b*x+c2*y(x))*diff(y(x),x)+a1+b1*x+b*y(x) = 0, y(x),'implicit')

Maple raw output

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