4.18.39 $4x{y}^{\prime }(x{)}^2+2x{y}^{\prime }(x)-y(x)=0$

ODE
$$4x{y}^{\prime }(x{)}^2+2x{y}^{\prime }(x)-y(x)=0$$ ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $y$

Mathematica ✓
cpu = 0.0808966 (sec), leaf count = 58

$$\{\{y(x)\to \frac{1}{4}{e}^{2c_1}(e^{2c_1}-2\sqrt{x})\},\{y(x)\to \frac{1}{4}{e}^{-4c_1}(1-2e^{2c_1}\sqrt{x})\}\}$$

Maple ✓
cpu = 0.021 (sec), leaf count = 29

$$\{y\left(x\right)=-\frac{x}{4},[x\left(\mathit{\_}\mathit{T}\right)=\frac{\mathit{\_}\mathit{C}\mathit{1}}{{\mathit{\_}\mathit{T}}^2},y\left(\mathit{\_}\mathit{T}\right)=\frac{(4\mathit{\_}\mathit{T}+2)\mathit{\_}\mathit{C}\mathit{1}}{\mathit{\_}\mathit{T}}]\}$$ Mathematica raw input

DSolve[-y[x] + 2*x*y'[x] + 4*x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (E^(2*C[1])*(E^(2*C[1]) - 2*Sqrt[x]))/4}, {y[x] -> (1 - 2*E^(2*C[1])*S
qrt[x])/(4*E^(4*C[1]))}}

Maple raw input

dsolve(4*x*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = -1/4*x, [x(_T) = 1/_T^2*_C1, y(_T) = (4*_T+2)*_C1/_T]