ODE types solvable by two quadratures
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 (http://planetmath.org/EquationYFx).
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
here the variables and integrating give the general integral of (3) in the form
The integration constant (http://planetmath.org/SolutionsOfOrdinaryDifferentialEquation) has an influence on the form of the integral curves, but only translates them in the direction of the -axis.
is equivalent (http://planetmath.org/Equivalent3) 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:
Accordingly we have in this case the general solution of the ODE (6):
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.
|Title||ODE types solvable by two quadratures|
|Date of creation||2015-03-20 17:04:58|
|Last modified on||2015-03-20 17:04:58|
|Last modified by||pahio (2872)|
|Synonym||second order ODE types solvable by quadratures|