The second order ordinary differential equation

(1)

If the right hand side of (1) contains at most one of the quantities $x$ , $y$ and $\frac{dy}{dx}$ , the general solution solution is obtained by two quadratures.

• The equation

  (2)

  
  is considered here.
  
• The equation

  (3)

  
  has as constant solutions all real roots of the equation $f(y) = 0$ . The other solutions can be gotten from the normal system

  (4)

  
  of (3). Dividing the equations (4) we get now $\frac{dz}{dy} = \frac{f(y)}{z}$ . By separation of variables and integration we may write $$\frac{z^2}{2} = \int\!f(y)\,dy +C_1,$$ whence the first equation of (4) reads $$\frac{dy}{dx} \,=\, \sqrt{2\!\int\!f(y)\,dy+C_1}.$$ Separating here the variables and integrating give the general integral of (3) in the form

  (5)

  
  The integration constant $C_1$ has an influence on the form of the integral curves, but $C_2$ only translates them in the direction of the $x$ -axis.
  
• The equation

  (6)

  
  is equivalent with the normal system

  (7)

  
  If the equation $f(z) = 0$ has real roots $z_1,\,z_2,\,\ldots$ , these satisfy the latter of the equations (7), and thus, according to the former of them, the differential equation (6) has the solutions $y := z_1x+C_1$ , $y := z_2x+C_2,\;\ldots$ .

  The other solutions of (6) are obtained by separating the variables and integrating:

  (8)

  
  If this antiderivative is expressible in closed form and if then the equation (8) can be solved for $z$ , we may write $$z \,=\, \frac{dy}{dx} \,=\, g(x\!-\!C).$$ Accordingly we have in this case the general solution of the ODE (6):

  (9)

  
  In other cases, we express also $y$ as a function of $z$ . By the chain rule, the normal system (7) yields $$\frac{dy}{dz} \,=\, \frac{dy}{dx}\cdot\frac{dx}{dz} \,=\, \frac{z}{f(z)},$$ whence $$y = \int\frac{z\,dz}{f(z)}+C'.$$ Thus the general solution of (6) reads now in a parametric form as

  (10)

  
  The equations 10 show that a translation of any integral curve yields another integral curve.