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 = 0.43438 (sec), leaf count = 153
\[\left \{\left \{y(x)\to -\sqrt {-i (\cosh (c_1)+\sinh (c_1)) (2 x-i \cosh (c_1)-i \sinh (c_1))}\right \},\left \{y(x)\to \sqrt {-i (\cosh (c_1)+\sinh (c_1)) (2 x-i \cosh (c_1)-i \sinh (c_1))}\right \},\left \{y(x)\to -\sqrt {i (\cosh (c_1)+\sinh (c_1)) (2 x+i \cosh (c_1)+i \sinh (c_1))}\right \},\left \{y(x)\to \sqrt {i (\cosh (c_1)+\sinh (c_1)) (2 x+i \cosh (c_1)+i \sinh (c_1))}\right \}\right \}\]
Maple ✓
cpu = 0.08 (sec), leaf count = 18
\[\left [-\textit {\_C1} +x +\sqrt {x^{2}+y \left (x \right )^{2}} = 0\right ]\] Mathematica raw input
DSolve[(x - Sqrt[x^2 + y[x]^2])*y'[x] == y[x],y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[(-I)*(2*x - I*Cosh[C[1]] - I*Sinh[C[1]])*(Cosh[C[1]] + Sinh[C[1]])]}, {y[x] -> Sqrt[(-I)*(2*x - I*Cosh[C[1]] - I*Sinh[C[1]])*(Cosh[C[1]] + Sinh[C[1]])]}, {y[x] -> -Sqrt[I*(2*x + I*Cosh[C[1]] + I*Sinh[C[1]])*(Cosh[C[1]] + Sinh[C[1]])]}, {y[x] -> Sqrt[I*(2*x + I*Cosh[C[1]] + I*Sinh[C[1]])*(Cosh[C[1]] + Sinh[C[1]])]}}
Maple raw input
dsolve((x-(x^2+y(x)^2)^(1/2))*diff(y(x),x) = y(x), y(x))
Maple raw output
[-_C1+x+(x^2+y(x)^2)^(1/2) = 0]