ODE No. 827

$$y^{\prime }(x)=\frac{x^{2}y(x)\sqrt{x^2+y(x{)}^2}+x^3(-\sqrt{x^2+y(x{)}^2})+y(x)}{x}$$ ✓ Mathematica : cpu = 0.278409 (sec), leaf count = 221

DSolve[Derivative[1][y][x] == (y[x] - x^3*Sqrt[x^2 + y[x]^2] + x^2*y[x]*Sqrt[x^2 + y[x]^2])/x,y[x],x]
 

$$\{\{y(x)\to \frac{x-2\sqrt{x^2{\tanh }^2\left(\frac{1}{3}(-\sqrt{2}{x}^3-3\sqrt{2}{c}_1)\right)-x^2{\tanh }^4\left(\frac{1}{3}(-\sqrt{2}{x}^3-3\sqrt{2}{c}_1)\right)}}{-1+2{\tanh }^2\left(\frac{1}{3}(-\sqrt{2}{x}^3-3\sqrt{2}{c}_1)\right)}\},\{y(x)\to \frac{x+2\sqrt{x^2{\tanh }^2\left(\frac{1}{3}(-\sqrt{2}{x}^3-3\sqrt{2}{c}_1)\right)-x^2{\tanh }^4\left(\frac{1}{3}(-\sqrt{2}{x}^3-3\sqrt{2}{c}_1)\right)}}{-1+2{\tanh }^2\left(\frac{1}{3}(-\sqrt{2}{x}^3-3\sqrt{2}{c}_1)\right)}\}\}$$ ✓ Maple : cpu = 0.192 (sec), leaf count = 49

dsolve(diff(y(x),x) = -(-y(x)+x^3*(y(x)^2+x^2)^(1/2)-x^2*(y(x)^2+x^2)^(1/2)*y(x))/x,y(x))
 

$$\ln \left(\frac{2x(\sqrt{2y{\left(x\right)}^2+2x^2}+y\left(x\right)+x)}{y\left(x\right)-x}\right)+\frac{\sqrt{2}{x}^3}{3}-\ln \left(x\right)-c_1=0$$