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

(i)

is the general solution to the homogeneous associated equation (H);

(ii)

is a particular solution to the equation (NH).

In conclusion, we deduce that in order to solve the nonhomogeneous equation (NH), we need to

Step 1: find the general solution to the homogeneous associated equation (H), say ;

Step 2: find a particular solution to the equation (NH), say ;

Step 3: write down the general solution to (NH) as

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