## ragman Group Title needed help with solving linear time invariant differential equations one year ago one year ago

1. ragman Group Title

obtain the solution x(t) of the differential equation

2. ragman Group Title

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3. UnkleRhaukus Group Title

$\ddot x+\omega_n^2x=t$

4. ragman Group Title

yup thats what i tried to write

5. UnkleRhaukus Group Title

$m^2+\omega_n^2=0$

6. ragman Group Title

would it be possible to explain that ?

7. UnkleRhaukus Group Title

the characteristic equation ?

8. ragman Group Title

so the first thing you would do is use the laplace transform on everything right im just not sure how to figure out how to do it for the w shaped variable

9. UnkleRhaukus Group Title

omega ? $$\omega_n^2$$ is a constant for constant n

10. UnkleRhaukus Group Title

im not sure how you are going to use Laplace if you dont have any initial values

11. ragman Group Title

|dw:1347266422040:dw|

12. ragman Group Title

i knew i was forgetting something

13. ragman Group Title

and yeah i meant omega

14. UnkleRhaukus Group Title

well if we are going to use laplace transform you can forget the characteristic equation

15. ragman Group Title

ok sounds good the only difficulty i face is when things like omega come up i just get confused as to how im supposed to find the laplace transform of it

16. UnkleRhaukus Group Title

its just a positive constant, if you want to set $\omega_n^2=A,\qquad A>0$

17. ragman Group Title

ok im sorta getting it so what would the laplace transform of this constant be

18. UnkleRhaukus Group Title

the laplace transform of a constant times a function is the constant times the transfer $\mathcal L\{αf(t)\} = α\mathcal L\{f(t)\}$ the laplace transform of one is one on the the transfer parameter $\mathcal L\{1\}(s)=\frac 1s$