How do I make a write a particular integral for f(t) in a 2nd order linear inhomogeneous ode when f(t) has two expressions?

[Al-Layth]

hello

I am trying to solve this 2nd order linear inhomogeneous ordinary differential equation
$$\frac{40}{17}x''-\frac{1}{17}x'+0.4x=6-\frac{123}{25}e^{-0.18t}$$
In order to solve this I need a particular integral for f(t) right( which is the rhs)
I know how to write the particular integral for the exponential and constant terms individually. But I don't know how I should do it when they're together like this.

So far I attempted to write the particular integral as:
$$\lambda_{1}+\lambda_{2}e^{kx}$$assuming I just need to sum the individual particular integrals for the constant and exponential terms just as they are summed in the ode.

is this the correct choice?

 

[topsquark]

I haven't done the problem but so long as the homogenous solution does not contain a $e^{-0.18t}$ then you can use $\lambda_1 + \lambda _2 e^{-0.18t}$ for the particular solution.

-Dan

 

[Al-Layth]

I haven't done the problem but so long as the homogenous solution does not contain a $e^{-0.18t}$ then you can use $\lambda_1 + \lambda _2 e^{-0.18t}$ for the particular solution.

-Dan

thanks
the homogeneous solution contains complex roots because the discriminant for the auxiliary is <0

 

[Deleted member 4993]

thanks
the homogeneous solution contains complex roots because the discriminant for the auxiliary is <0

You should make sure whether "A*e^(-0.18t)" satisfy the homogeneous equation or not - prior to deciding on the form of the particular solution. If it does satisfy the homogeneous equation, then you need to think about it (almost similar but not the same - as resonance in simple harmonic motion - where the frequency associated with forcing function was equal to the natural frequency).

 

[Al-Layth]

You should make sure whether "A*e^(-0.18t)" satisfy the homogeneous equation or not - prior to deciding on the form of the particular solution. If it does satisfy the homogeneous equation, then you need to think about it (almost similar but not the same - as resonance in simple harmonic motion - where the frequency associated with forcing function was equal to the natural frequency).

thanks
one of the expressions I have in my complementary function is
$$e^{\frac{1}{80}x}$$
But the expression in my Forcing function is:
$$-\frac{123}{25}e^{-0.18t}$$
do i still need to change the p.i? even thought the term above e is different?

I could check by subbing in of course but I want to know how I can tell without having to do that

Edit: turns out I have to sub my p.I regardless, to determine its constants' values