4.17.18 $y^{\prime }(x{)}^2-(4y(x)+1)y^{\prime }(x)+y(x)(4y(x)+1)=0$

ODE
$$y^{\prime }(x{)}^2-(4y(x)+1)y^{\prime }(x)+y(x)(4y(x)+1)=0$$ ODE Classification

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.0499261 (sec), leaf count = 55

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

Maple ✓
cpu = 0.048 (sec), leaf count = 49

$$\{x-\frac{\ln \left(y\left(x\right)\right)}{2}-\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\sqrt{1+4y\left(x\right)}\right)-\mathit{\_}\mathit{C}\mathit{1}=0,x-\frac{\ln \left(y\left(x\right)\right)}{2}+\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\sqrt{1+4y\left(x\right)}\right)-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=-\frac{1}{4}\}$$ Mathematica raw input

DSolve[y[x]*(1 + 4*y[x]) - (1 + 4*y[x])*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(diff(y(x),x)^2-(1+4*y(x))*diff(y(x),x)+(1+4*y(x))*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = -1/4, x-1/2*ln(y(x))+arctanh((1+4*y(x))^(1/2))-_C1 = 0, x-1/2*ln(y(x))-ar
ctanh((1+4*y(x))^(1/2))-_C1 = 0