4.16.42 \((a+x) y'(x)+y'(x)^2-y(x)=0\)

ODE
\[ (a+x) y'(x)+y'(x)^2-y(x)=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _Clairaut]

Book solution method
Clairaut’s equation and related types, main form

Mathematica
cpu = 0.00254642 (sec), leaf count = 13

\[\left \{\left \{y(x)\to c_1 \left (a+c_1+x\right )\right \}\right \}\]

Maple
cpu = 0.019 (sec), leaf count = 20

\[ \left \{ y \relax (x ) ={\it \_C1}\, \left ({\it \_C1}+a+x \right ) ,y \relax (x ) =-{\frac { \left (a+x \right ) ^{2}}{4}} \right \} \] Mathematica raw input

DSolve[-y[x] + (a + x)*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]*(a + x + C[1])}}

Maple raw input

dsolve(diff(y(x),x)^2+(a+x)*diff(y(x),x)-y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = -1/4*(a+x)^2, y(x) = _C1*(_C1+a+x)