4.15.33 $(x-\sqrt{x^2+y(x{)}^2})y^{\prime }(x)=y(x)$

ODE
$$(x-\sqrt{x^2+y(x{)}^2})y^{\prime }(x)=y(x)$$ ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.0447243 (sec), leaf count = 52

$$\{\{y(x)\to -e^{\frac{c_1}{2}}\sqrt{e^{c_1}-2x}\},\{y(x)\to e^{\frac{c_1}{2}}\sqrt{e^{c_1}-2x}\}\}$$

Maple ✓
cpu = 0.062 (sec), leaf count = 18

$$\{-\mathit{\_}\mathit{C}\mathit{1}+x+\sqrt{x^2+{\left(y\left(x\right)\right)}^2}=0\}$$ 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