4.15.28 \(\left (\sqrt {y(x)+x}+1\right ) y'(x)+1=0\)

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 = 0.0324476 (sec), leaf count = 39

\[\left \{\left \{y(x)\to -2 \sqrt {c_1+x+1}+c_1+2\right \},\left \{y(x)\to 2 \sqrt {c_1+x+1}+c_1+2\right \}\right \}\]

Maple
cpu = 0.024 (sec), leaf count = 19

\[ \left \{ -2\,\sqrt {x+y \relax (x ) }-y \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

-2*(x+y(x))^(1/2)-y(x)-_C1 = 0