4.18.14 \((a-y(x)) y'(x)+b+x y'(x)^2=0\)

ODE
\[ (a-y(x)) y'(x)+b+x y'(x)^2=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)

Mathematica
cpu = 0.702642 (sec), leaf count = 212

\[\left \{\left \{y(x)\to -\frac {-a b+a \sinh \left (2 c_1\right )+a \cosh \left (2 c_1\right )+b^2+\sqrt {b \left (\sinh \left (4 c_1\right )+\cosh \left (4 c_1\right )\right ) \left ((-b+x-1) \sinh \left (c_1\right )+(b-x-1) \cosh \left (c_1\right )\right ){}^2}-b \sinh \left (2 c_1\right )-b \cosh \left (2 c_1\right )+b x}{b-\sinh \left (2 c_1\right )-\cosh \left (2 c_1\right )}\right \},\left \{y(x)\to \frac {a b-a \sinh \left (2 c_1\right )-a \cosh \left (2 c_1\right )-b^2+\sqrt {b \left (\sinh \left (4 c_1\right )+\cosh \left (4 c_1\right )\right ) \left ((-b+x-1) \sinh \left (c_1\right )+(b-x-1) \cosh \left (c_1\right )\right ){}^2}+b \sinh \left (2 c_1\right )+b \cosh \left (2 c_1\right )-b x}{b-\sinh \left (2 c_1\right )-\cosh \left (2 c_1\right )}\right \}\right \}\]

Maple
cpu = 0.027 (sec), leaf count = 37

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-2\,ay \relax (x ) +{a}^{2}-4\,bx=0,y \relax (x ) ={\frac {x{{\it \_C1}}^{2}+a{\it \_C1}+b}{{\it \_C1}}} \right \} \] Mathematica raw input

DSolve[b + (a - y[x])*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((-(a*b) + b^2 + b*x + a*Cosh[2*C[1]] - b*Cosh[2*C[1]] + a*Sinh[2*C[1
]] - b*Sinh[2*C[1]] + Sqrt[b*((-1 + b - x)*Cosh[C[1]] + (-1 - b + x)*Sinh[C[1]])
^2*(Cosh[4*C[1]] + Sinh[4*C[1]])])/(b - Cosh[2*C[1]] - Sinh[2*C[1]]))}, {y[x] ->
 (a*b - b^2 - b*x - a*Cosh[2*C[1]] + b*Cosh[2*C[1]] - a*Sinh[2*C[1]] + b*Sinh[2*
C[1]] + Sqrt[b*((-1 + b - x)*Cosh[C[1]] + (-1 - b + x)*Sinh[C[1]])^2*(Cosh[4*C[1
]] + Sinh[4*C[1]])])/(b - Cosh[2*C[1]] - Sinh[2*C[1]])}}

Maple raw input

dsolve(x*diff(y(x),x)^2+(a-y(x))*diff(y(x),x)+b = 0, y(x),'implicit')

Maple raw output

y(x)^2-2*a*y(x)+a^2-4*b*x = 0, y(x) = (_C1^2*x+_C1*a+b)/_C1