Differential equations...

Question

anonymous · Answer

this isnt even a question

anonymous · Answer

I'm busy typing it...

anonymous · Answer

don't worry, I have patience! :)

anonymous · Answer

oh im sorry taljaards |dw:1368650170667:dw|

anonymous · Answer

a) Solve the initial value problem using Laplace Transformation:$$y^n+16y=25e^{-3t}\y(0)=y'(0)=0$$

b) Let 0 < a < b be given.  Write the function f in terms of the Heaviside function and calculate the Laplace Transformation thereof$$f(t)=0; 0<a\f(t)=e^{-t}; a<t<b\f(t)=0;t>b$$

anonymous · Answer

wow that's hard, im so sorry I cant help, I haven't studied this yet!! :(

anonymous · Answer

Yip I'm struggling as well!  It is preparation for our test tomorrow, but I actually don't know what to do...

anonymous · Answer

$$y''+16y=25e^{-3t}$$
Taking the Laplace transform of both sides (I'm using the notation $F(s)=\mathscr{L}\left\{f(t)\right\}$):
$$s^2Y(s)-s~y(0) - y'(0)+16~Y(s)=\frac{25}{s+3}\
(s^2+16)Y(s)=\frac{25}{s+3}\
Y(s)=\frac{25}{(s+3)(s^2+16)}$$
Now for some partial fraction decomposition; solve for A, B, and C:
$$\frac{25}{(s+3)(s^2+16)}=\frac{A}{s+3}+\frac{Bs+C}{s^2+16}$$

anonymous · Answer

As for the second question; you're given the function
$$f(t)=\begin{cases}e^{-t}&\text{for }a<t<b\
0&\text{otherwise}\end{cases}$$
Here's a sketch of the function for some arbitrary $a$ and $b$:
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