ODE
\[ \left (\sqrt {y(x)+x}+1\right ) y'(x)+1=0 \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
Change of Variable, new dependent variable
Mathematica ✓
cpu = 0.302526 (sec), leaf count = 39
\[\left \{\left \{y(x)\to -2 \sqrt {x+1+c_1}+2+c_1\right \},\left \{y(x)\to 2 \sqrt {x+1+c_1}+2+c_1\right \}\right \}\]
Maple ✓
cpu = 0.034 (sec), leaf count = 19
\[\left [-y \left (x \right )-2 \sqrt {x +y \left (x \right )}-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[1 + (1 + Sqrt[x + y[x]])*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> 2 + C[1] - 2*Sqrt[1 + x + C[1]]}, {y[x] -> 2 + C[1] + 2*Sqrt[1 + x + C[1]]}}
Maple raw input
dsolve((1+(x+y(x))^(1/2))*diff(y(x),x)+1 = 0, y(x))
Maple raw output
[-y(x)-2*(x+y(x))^(1/2)-_C1 = 0]