Linear Second Order Differential Equations
Linear Second Order Differential Equations
A linear second order differential equations is written as
When d(x) = 0, the equation is called homogeneous, otherwise it is called nonhomogeneous. To a nonhomogeneous equation
,
we associate the so called associated homogeneous equation
For the study of these equations we consider the explicit ones given by
where p(x) = b(x)/a(x), q(x) = c(x)/a(x) and g(x) = d(x)/a(x). If p(x), q(x) and g(x) are defined and continuous on the interval I, then the IVP
,
where 
and 
are arbitrary numbers, has a
unique solution defined on I.
Main result: The general solution to
the equation (NH) is given by
where
, 
is the general solution to the homogeneous associated
equation (H);
is a particular solution to the equation (NH).
In conclusion, we deduce that in order to solve the nonhomogeneous equation (NH), we need to
;
;
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Author: Mohamed Amine Khamsi