The easiest way to remember that homogeneous equation is one of two solutions to a second order ordinary differential equation is that even if the right hand side is zero:
y'' + b(x) y' + c(x) = 0,
this equation will still yield a solution y(x). The second solution appears, in addition to the homogeneous, if the right hand side is not zero:
y'' + b(x) y' + c(x) = r(x).
Physically, the homogeneous solution represents the internal response of a mechanical system, while r(x) represents an external force that drives the system. The driving force r(x) will produce a particular solution of the system in response to the external driving force. This is - in my opinion - a more practical way to think about homogeneous and particular solutions.
It is also important to point out that the initial conditions only influence the homogeneous solution. This feature helps to tell apart the homogeneous and the particular in a first order equation when we use an integration factor to solve it. The homogeneous solution is simply one that has the integration constant. This aspect is important to remember since we often think that homogeneous and particular solutions appear only in second and higher order equations.
There is a theorem in differential equation that describes the above more succinctly. I could not recall it today in lectures, and it motivated me to find an alternative way to describe homogeneous and particular solutions of second order equation.
y'' + b(x) y' + c(x) = 0,
this equation will still yield a solution y(x). The second solution appears, in addition to the homogeneous, if the right hand side is not zero:
y'' + b(x) y' + c(x) = r(x).
Physically, the homogeneous solution represents the internal response of a mechanical system, while r(x) represents an external force that drives the system. The driving force r(x) will produce a particular solution of the system in response to the external driving force. This is - in my opinion - a more practical way to think about homogeneous and particular solutions.
It is also important to point out that the initial conditions only influence the homogeneous solution. This feature helps to tell apart the homogeneous and the particular in a first order equation when we use an integration factor to solve it. The homogeneous solution is simply one that has the integration constant. This aspect is important to remember since we often think that homogeneous and particular solutions appear only in second and higher order equations.
There is a theorem in differential equation that describes the above more succinctly. I could not recall it today in lectures, and it motivated me to find an alternative way to describe homogeneous and particular solutions of second order equation.
No comments:
Post a Comment