## anonymous 3 years ago needed help with solving linear time invariant differential equations

1. anonymous

obtain the solution x(t) of the differential equation

2. anonymous

|dw:1347265100080:dw|

3. UnkleRhaukus

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

4. anonymous

yup thats what i tried to write

5. UnkleRhaukus

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

6. anonymous

would it be possible to explain that ?

7. UnkleRhaukus

the characteristic equation ?

8. anonymous

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

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

10. UnkleRhaukus

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

11. anonymous

|dw:1347266422040:dw|

12. anonymous

i knew i was forgetting something

13. anonymous

and yeah i meant omega

14. UnkleRhaukus

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

15. anonymous

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

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

17. anonymous

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

18. UnkleRhaukus

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 $\mathcal L\{1\}(s)=\frac 1s$