4.18.16 $(a-y(x)+x)y^{\prime }(x)+x{y}^{\prime }(x{)}^2-y(x)=0$

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

[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

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

Mathematica ✓
cpu = 0.877841 (sec), leaf count = 145

$$\{\{y(x)\to -\frac{-3a^2+\sqrt{-a{e}^{4c_1}(a+e^{4c_1}+4x){}^2}-3a{e}^{4c_1}+6ax+2e^{4c_1}x}{2(a+e^{4c_1})}\},\{y(x)\to \frac{3a^2+3a(e^{4c_1}-2x)+\sqrt{-a{e}^{4c_1}(a+e^{4c_1}+4x){}^2}-2e^{4c_1}x}{2(a+e^{4c_1})}\}\}$$

Maple ✓
cpu = 0.038 (sec), leaf count = 39

$$\{{\left(y\left(x\right)\right)}^2+(-2a+2x)y\left(x\right)+{(a+x)}^2=0,y\left(x\right)=\frac{\mathit{\_}\mathit{C}\mathit{1}(\mathit{\_}\mathit{C}\mathit{1}x+a+x)}{\mathit{\_}\mathit{C}\mathit{1}+1}\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -(-3*a^2 - 3*a*E^(4*C[1]) + 6*a*x + 2*E^(4*C[1])*x + Sqrt[-(a*E^(4*C[1
])*(a + E^(4*C[1]) + 4*x)^2)])/(2*(a + E^(4*C[1])))}, {y[x] -> (3*a^2 + 3*a*(E^(
4*C[1]) - 2*x) - 2*E^(4*C[1])*x + Sqrt[-(a*E^(4*C[1])*(a + E^(4*C[1]) + 4*x)^2)]
)/(2*(a + E^(4*C[1])))}}

Maple raw input

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

Maple raw output

y(x)^2+(-2*a+2*x)*y(x)+(a+x)^2 = 0, y(x) = _C1*(_C1*x+a+x)/(_C1+1)