Simple linear 1st order but hard to figure out

[Levido]

Hi all, I’ve tried this question a few times and ill post my working as far as I’ve been able to get it.

Here’s the question:



And here’s how “far” I managed to get



From the back I know that my differential equation is set up right but the form at the end only looks a bit like what it needs to be ( 2 instead of sqrt(2) and constant should be 0)

Thank you for any help you give and your time

 

[lex]

[MATH] \begin{align*} \sin(ct) - \cos(ct) & = \sqrt{2} \left(\frac{1}{\sqrt{2}} \sin(ct) - \frac{1}{\sqrt{2}} \cos(ct)\right)\\ & = \sqrt{2}(\sin(ct) \cos(\frac{\pi}{4}) - \cos(ct)\sin(\frac{\pi}{4}))\\ & = \sqrt{2}(\sin(ct-\frac{\pi}{4}))\\ \end{align*} [/MATH]

 

[Levido]

[MATH] \begin{align*} \sin(ct) - \cos(ct) & = \sqrt{2} \left(\frac{1}{\sqrt{2}} \sin(ct) - \frac{1}{\sqrt{2}} \cos(ct)\right)\\ & = \sqrt{2}(\sin(ct) \cos(\frac{\pi}{4}) - \cos(ct)\sin(\frac{\pi}{4}))\\ & = \sqrt{2}(\sin(ct-\frac{\pi}{4}))\\ \end{align*} [/MATH]

Thanks Lex that’s very cool

Now all I need assistance with is the constant part but I have a feeling that that’s due to an algebra error

 

[lex]

Note - we do not know where the cat is when [MATH]t=0[/MATH].

So [MATH] \boxed{x=a + \frac{b}{\sqrt{2}} \sin(ct-\frac{\pi}{4}) +d e^{-ct}}[/MATH]The wording of the question is strange: Show that, after some time, [MATH]x[/MATH] is approximately equal to [MATH]a + \frac{b}{\sqrt{2}} \sin(ct-\frac{\pi}{4})[/MATH]which indicates that they do not want us to find [MATH]d[/MATH], but only observe what happens to [MATH]x[/MATH] as [MATH]t[/MATH] gets large.

I.e. what happens to the final term of [MATH] \hspace4ex x=a + \frac{b}{\sqrt{2}} \sin(ct-\frac{\pi}{4}) +d e^{-ct} \hspace4ex \text{ as } t \text{ becomes large?}[/MATH]What does that mean for the equation for [MATH]x[/MATH], as [MATH]t[/MATH] becomes large?

 

[Levido]

Note - we do not know where the cat is when [MATH]t=0[/MATH].

1) Good spot, I’m tempted to plot this on Desmos and change the starting position of the cat

Thanks Lex for your help

 

[lex]

Grand. Hopefully it is now clear why [MATH]x \approx a + \frac{b}{\sqrt{2}} \sin(ct-\frac{\pi}{4}) [/MATH] "after some time".

 

[Levido]

Grand. Hopefully it is now clear why [MATH]x \approx a + \frac{b}{\sqrt{2}} \sin(ct-\frac{\pi}{4}) [/MATH] "after some time".

Yeah, from pre calculus you can imagine it as x—>+infinity

Thanks for your help last night Lex

 

[lex]

As [MATH]\hspace1ex t\rightarrow \infty [/MATH]No problem. Glad it helped. You had all the hard work done very well.