4.15.33 \(\left (x-\sqrt {x^2+y(x)^2}\right ) y'(x)=y(x)\)

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

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