why it is particular solution + general solution?

[jsboy]

Hi,

As to forced frequcncy analysis, theoretically it is to solve 2nd differential equation and the solutoin would be summation of general solution and particular solution, which I can understand exactly as this behavior is just the sum of transient and steady phonenomenon. However apart from vibration analysis and think about mathmatics only, is general solution plus particular solution always the solution for any 2nd differential equation? Only general solution can be the answer to some 2nd differential equation?

 

[davefitz]

The general solution toa DE is for homogenous boundary conditions, and contains the natural basis functions for which one can compose the furhter solution for particular problems . The specific or particular solution is for non-homogenous boundary conditions. One can form the solution of the particular problem using a summation of the natural basis functions found from the homogenous problem.

 

[electricpete]

To my memory the issue is not one of boundary conditions but forcing functions.

Homogenous solution applies to the unforced system.

Particular solution applies to the forced system.

The complete solution is the sum of the homogenous solution(s) and the particular solution. There are undetermined coefficients of the complete solution that will be evaluated based on the boundary conditions.

Homogenous solution is all you need if you wish to only study the unforced response of a system (starting at some non-zero initial condition).

But it won't tell you what happens when you apply a forcing funciton... that requires the particular solution.

Hope I said it right.... if not hope someone will correct me.

 

[jsboy]

I think electricpete is correct from the point view of vibration analysis. But let's say we have some 2nd differential equation to solve, ex) x"+2x'+5x=f(x), and forget about whether this is vibration study or not, but it is just a mathematical equation. Then my question is that the same rule can be applied and the solution to this is summation of general solution (f=0) and particular solution(f not 0)? We can not simply solve the equation above only one time with f(x) not being zero?

 

[electricpete]

The idea of homogenous and particular are not unique to vib analysis. In general you will need both to satisfy all of your boundary conditions.