ODE No. 386

$$a{x}^3{y}^{\prime }(x)-2a{x}^{2}y(x)+y^{\prime }(x{)}^2=0$$ ✓ Mathematica : cpu = 0.175172 (sec), leaf count = 119

DSolve[-2*a*x^2*y[x] + a*x^3*Derivative[1][y][x] + Derivative[1][y][x]^2 == 0,y[x],x]
 

$$\{\{y(x)\to \frac{1}{2}(\cosh (2c_1)+\sinh (2c_1))(-\sqrt{2}\sqrt{a}{x}^2+2\cosh (2c_1)+2\sinh (2c_1))\},\{y(x)\to \frac{\sqrt{a}{x}^2\cosh (2c_1)}{\sqrt{2}}+\frac{\sqrt{a}{x}^2\sinh (2c_1)}{\sqrt{2}}+{\cosh }^2(2c_1)+{\sinh }^2(2c_1)+2\sinh (2c_1)\cosh (2c_1)\}\}$$ ✓ Maple : cpu = 0.784 (sec), leaf count = 27

dsolve(diff(y(x),x)^2+a*x^3*diff(y(x),x)-2*a*x^2*y(x) = 0,y(x))
 

$$y\left(x\right)=-\frac{a{x}^4}{8}$$