4.18.14 $(a-y(x))y^{\prime }(x)+b+x{y}^{\prime }(x{)}^2=0$

ODE
$$(a-y(x))y^{\prime }(x)+b+x{y}^{\prime }(x{)}^2=0$$ ODE Classification

[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

Book solution method
Clairaut’s equation and related types, $f(y-x{y}^{\prime },y^{\prime })=0$

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

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

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

$$\{{\left(y\left(x\right)\right)}^2-2ay\left(x\right)+a^2-4bx=0,y\left(x\right)=\frac{x{\mathit{\_}\mathit{C}\mathit{1}}^2+a\mathit{\_}\mathit{C}\mathit{1}+b}{\mathit{\_}\mathit{C}\mathit{1}}\}$$ 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