4.3.38 \(y'(x)=f(a+b x+c y(x))\)

ODE
\[ y'(x)=f(a+b x+c y(x)) \] ODE Classification

[[_homogeneous, `class C`], _dAlembert]

Book solution method
Equation linear in the variables, \(y'(x)=f(a+b x+ c y(x))\)

Mathematica
cpu = 11.4208 (sec), leaf count = 126

\[\text {Solve}\left [c_1=\int _1^{y(x)} -\frac {(c f(c K[2]+a+b x)+b) \int _1^x \frac {b c^2 f'(b K[1]+c K[2]+a)}{(c f(b K[1]+c K[2]+a)+b)^2} \, dK[1]+c}{c f(c K[2]+a+b x)+b} \, dK[2]+\int _1^x \frac {c f(b K[1]+a+c y(x))}{c f(b K[1]+a+c y(x))+b} \, dK[1],y(x)\right ]\]

Maple
cpu = 0.032 (sec), leaf count = 35

\[ \left \{ -\int ^{y \relax (x ) +{\frac {bx}{c}}}\! \left (f \left ({\it \_a}\,c+a \right ) c+b \right ) ^{-1}{d{\it \_a}}c+x-{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y'[x] == f[a + b*x + c*y[x]],y[x],x]

Mathematica raw output

Solve[C[1] == Integrate[(c*f[a + b*K[1] + c*y[x]])/(b + c*f[a + b*K[1] + c*y[x]]
), {K[1], 1, x}] + Integrate[-((c + (b + c*f[a + b*x + c*K[2]])*Integrate[(b*c^2
*Derivative[1][f][a + b*K[1] + c*K[2]])/(b + c*f[a + b*K[1] + c*K[2]])^2, {K[1],
 1, x}])/(b + c*f[a + b*x + c*K[2]])), {K[2], 1, y[x]}], y[x]]

Maple raw input

dsolve(diff(y(x),x) = f(a+b*x+c*y(x)), y(x),'implicit')

Maple raw output

-Intat(1/(f(_a*c+a)*c+b),_a = y(x)+b/c*x)*c+x-_C1 = 0