ODE No. 413

$$-x^2+x{y}^{\prime }(x{)}^2+y(x)y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.111246 (sec), leaf count = 673

DSolve[-x^2 + y[x]*Derivative[1][y][x] + x*Derivative[1][y][x]^2 == 0,y[x],x]
 

$$\{\text{Solve}[\int (\frac{4\sqrt{4x^3+y(x{)}^2}{x}^2}{5y(x)(4x^3-15y(x{)}^2)}+\frac{16x^2}{5(4x^3-15y(x{)}^2)}-\frac{\sqrt{4x^3+y(x{)}^2}}{5y(x)x}+\frac{1}{5x})dx+\int (\frac{8y(x)}{15y(x{)}^2-4x^3}-\int (-\frac{4\sqrt{4x^3+y(x{)}^2}{x}^2}{5y(x{)}^2(4x^3-15y(x{)}^2)}+\frac{24\sqrt{4x^3+y(x{)}^2}{x}^2}{{(4x^3-15y(x{)}^2)}^2}+\frac{4x^2}{5(4x^3-15y(x{)}^2)\sqrt{4x^3+y(x{)}^2}}+\frac{96y(x)x^2}{{(4x^3-15y(x{)}^2)}^2}+\frac{\sqrt{4x^3+y(x{)}^2}}{5y(x{)}^{2}x}-\frac{1}{5\sqrt{4x^3+y(x{)}^2}x})dx+\frac{2\sqrt{4x^3+y(x{)}^2}}{15y(x{)}^2-4x^3})dy(x)=c_1,y(x)],\text{Solve}[\int (-\frac{4\sqrt{4x^3+y(x{)}^2}{x}^2}{5y(x)(4x^3-15y(x{)}^2)}+\frac{16x^2}{5(4x^3-15y(x{)}^2)}+\frac{\sqrt{4x^3+y(x{)}^2}}{5y(x)x}+\frac{1}{5x})dx+\int (\frac{8y(x)}{15y(x{)}^2-4x^3}-\int (\frac{4\sqrt{4x^3+y(x{)}^2}{x}^2}{5y(x{)}^2(4x^3-15y(x{)}^2)}-\frac{24\sqrt{4x^3+y(x{)}^2}{x}^2}{{(4x^3-15y(x{)}^2)}^2}-\frac{4x^2}{5(4x^3-15y(x{)}^2)\sqrt{4x^3+y(x{)}^2}}+\frac{96y(x)x^2}{{(4x^3-15y(x{)}^2)}^2}-\frac{\sqrt{4x^3+y(x{)}^2}}{5y(x{)}^{2}x}+\frac{1}{5\sqrt{4x^3+y(x{)}^2}x})dx-\frac{2\sqrt{4x^3+y(x{)}^2}}{15y(x{)}^2-4x^3})dy(x)=c_1,y(x)]\}$$ ✓ Maple : cpu = 0.173 (sec), leaf count = 269

dsolve(x*diff(y(x),x)^2+y(x)*diff(y(x),x)-x^2 = 0,y(x))
 

$${\int }_{\mathit{\text{\_b}}}^x\frac{-y\left(x\right)+\sqrt{4{\mathit{\text{\_a}}}^3+y{\left(x\right)}^2}}{\mathit{\text{\_a}}(4y\left(x\right)-\sqrt{4{\mathit{\text{\_a}}}^3+y{\left(x\right)}^2})}d\mathit{\text{\_a}}+{\int }_{}^{y\left(x\right)}\frac{-2+(48\mathit{\text{\_f}}-12\sqrt{4x^3+{\mathit{\text{\_f}}}^2})\left({\int }_{\mathit{\text{\_b}}}^x\frac{{\mathit{\text{\_a}}}^2}{{(-4\mathit{\text{\_f}}+\sqrt{4{\mathit{\text{\_a}}}^3+{\mathit{\text{\_f}}}^2})}^2\sqrt{4{\mathit{\text{\_a}}}^3+{\mathit{\text{\_f}}}^2}}d\mathit{\text{\_a}}\right)}{4\mathit{\text{\_f}}-\sqrt{4x^3+{\mathit{\text{\_f}}}^2}}d\mathit{\text{\_f}}+c_1=0$$