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 = 0.197723 (sec), leaf count = 52
\[\left \{\left \{y(x)\to x-c_1 \sqrt {x+1}+1+\frac {c_1{}^2}{4}\right \},\left \{y(x)\to x+c_1 \sqrt {x+1}+1+\frac {c_1{}^2}{4}\right \}\right \}\]
Maple ✓
cpu = 0.076 (sec), leaf count = 99
\[\left [y \left (x \right ) = 0, y \left (x \right ) = \frac {x \left (x +1+\sqrt {x \textit {\_C1} +\textit {\_C1} +x +1}\right )^{2}}{\left (x +1\right )^{2}}+\frac {\left (x +1+\sqrt {x \textit {\_C1} +\textit {\_C1} +x +1}\right )^{2}}{\left (x +1\right )^{2}}, y \left (x \right ) = \frac {x \left (-x -1+\sqrt {x \textit {\_C1} +\textit {\_C1} +x +1}\right )^{2}}{\left (x +1\right )^{2}}+\frac {\left (-x -1+\sqrt {x \textit {\_C1} +\textit {\_C1} +x +1}\right )^{2}}{\left (x +1\right )^{2}}\right ]\] 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((x+1)*diff(y(x),x)^2 = y(x), y(x))
Maple raw output
[y(x) = 0, y(x) = x*(x+1+(_C1*x+_C1+x+1)^(1/2))^2/(x+1)^2+(x+1+(_C1*x+_C1+x+1)^(1/2))^2/(x+1)^2, y(x) = x*(-x-1+(_C1*x+_C1+x+1)^(1/2))^2/(x+1)^2+(-x-1+(_C1*x+_C1+x+1)^(1/2))^2/(x+1)^2]