ODE
\[ \left (x-\sqrt {x^2+y(x)^2}\right ) y'(x)=y(x) \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 0.0447243 (sec), leaf count = 52
\[\left \{\left \{y(x)\to -e^{\frac {c_1}{2}} \sqrt {e^{c_1}-2 x}\right \},\left \{y(x)\to e^{\frac {c_1}{2}} \sqrt {e^{c_1}-2 x}\right \}\right \}\]
Maple ✓
cpu = 0.062 (sec), leaf count = 18
\[ \left \{ -{\it \_C1}+x+\sqrt {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}=0 \right \} \] Mathematica raw input
DSolve[(x - Sqrt[x^2 + y[x]^2])*y'[x] == y[x],y[x],x]
Mathematica raw output
{{y[x] -> -(E^(C[1]/2)*Sqrt[E^C[1] - 2*x])}, {y[x] -> E^(C[1]/2)*Sqrt[E^C[1] - 2*x]}}
Maple raw input
dsolve((x-(x^2+y(x)^2)^(1/2))*diff(y(x),x) = y(x), y(x),'implicit')
Maple raw output
-_C1+x+(x^2+y(x)^2)^(1/2) = 0