2nd order non-homogeneous differential equation

Question

I been thinking alot about physical meaning of mathematical equations. But i am unable to find it. I hope you guys solve it for me. Why do 2nd order non-homogeneous differential equations have particular solution and complementary solution. Physically what does it mean??

Answer 1

The physical meaning of the particular solution and complementary solutions have something to do with the transient solution and steady sate solution.

The transient solution is the part which goes to zero very fast and it is the complementary solution which decays exponentially with time.

The steady state solution is the part which does not go away and it is the particular solution.

For example for $$ y''+3y'+2y= x$$

We have$$ y= C_1e^{-x}+C_2 e^{-2x}+x/2-3/4$$

The transient solution is $$ y_c= C_1e^{-x}+C_2 e^{-2x}$$

and the steady state solution is $$y_p=x/2-3/4$$