4.18.30 $(x+1)y^{\prime }(x{)}^2=y(x)$

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

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

Book solution method
No Missing Variables ODE, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.0186776 (sec), leaf count = 52

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

Maple ✓
cpu = 0.02 (sec), leaf count = 40

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

y(x) = 0, [x(_T) = 1/(_T-1)^2*(-_T^2+_C1+2*_T), y(_T) = _T^2*(_C1+1)/(_T-1)^2]