ODE
\[ (x+1) y'(x)^2=y(x) \] ODE Classification
[[_homogeneous, `class C`], _rational, _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y'\)
Mathematica ✓
cpu = 0.0186776 (sec), leaf count = 52
\[\left \{\left \{y(x)\to -c_1 \sqrt {x+1}+\frac {c_1^2}{4}+x+1\right \},\left \{y(x)\to c_1 \sqrt {x+1}+\frac {c_1^2}{4}+x+1\right \}\right \}\]
Maple ✓
cpu = 0.02 (sec), leaf count = 40
\[ \left \{ y \left ( x \right ) =0,[x \left ( {\it \_T} \right ) ={\frac {-{{\it \_T}}^{2}+{\it \_C1}+2\,{\it \_T}}{ \left ( {\it \_T}-1 \right ) ^{2}}},y \left ( {\it \_T} \right ) ={\frac {{{\it \_T}}^{2} \left ( {\it \_C1}+1 \right ) }{ \left ( {\it \_T}-1 \right ) ^{2}}}] \right \} \] 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]