The second order ordinary differential equation
may in certain special cases be solved by using two quadratures, sometimes also by reduction to a first order differential equation and a quadrature.
If the right hand side of (1) contains at most one of the quantities , and , the general solution solution is obtained by two quadratures.
is considered here.
has as constant solutions all real roots of the equation . The other solutions can be gotten from the normal system
of (3). Dividing the equations (4) we get now . By separation of variables and integration we may write
whence the first equation of (4) reads
Separating here the variables and integrating give the general integral of (3) in the form
The integration constant has an influence on the form of the integral curves, but only translates them in the direction of the -axis.
is equivalent with the normal system
If the equation has real roots , these satisfy the latter of the equations (7), and thus, according to the former of them, the differential equation (6) has the solutions , .
The other solutions of (6) are obtained by separating the variables and integrating:
If this antiderivative is expressible in closed form and if then the equation (8) can be solved for , we may write
Accordingly we have in this case the general solution of the ODE (6):
In other cases, we express also as a function of . By the chain rule, the normal system (7) yields
Thus the general solution of (6) reads now in a parametric form as
The equations 10 show that a translation of any integral curve yields another integral curve.