- #1
lampCable
- 22
- 1
Homework Statement
Given the IVP
[tex]
\ddot{r}+\dot{r}+20r=0\\
r(0) = 0.8\\
\dot{r}(0) = 0
[/tex]
for the length of an oscillating spring (damped), we find that the general solution is
[tex]
r=e^{-0.5t}[0.8\cos(\sqrt{19.75}t)+\frac{0.4}{\sqrt{19.75}}\sin(\sqrt{19.75}t)]
[/tex]
and I wish to find the curve bounding the upper part of the solution, i.e. the amplitude of the oscillations.
Homework Equations
The Attempt at a Solution
I understand that if
[tex]
r=e^{-0.5t}0.8\cos(\sqrt{19.75}t)
[/tex]
for example, then the amplitude would be obtained by removing the cosine factor, since [itex]e^{-0.5t}0.8[/itex] correspond to the amplitude and [itex]\cos(\sqrt{19.75}t)[/itex] corresponds to how fast the spring is oscillating. But how does one do when there are two oscillating terms as above?