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An ODE / f(w) where the ten f(v) can cpleSU fuction of the ratio OMly. Mid homogencous Such eqpatious CAI awzys be tratusfonued JO spatrable (puations bw cutulge of...

Question

An ODE / f(w) where the ten f(v) can cpleSU fuction of the ratio OMly. Mid homogencous Such eqpatious CAI awzys be tratusfonued JO spatrable (puations bw cutulge of the variabk.

An ODE / f(w) where the ten f(v) can cpleSU fuction of the ratio OMly. Mid homogencous Such eqpatious CAI awzys be tratusfonued JO spatrable (puations bw cutulge of the variabk.



Answers

The heavily damped simple harmonic system of Figure $2.2$ is displaced a distance $F$ from its equilibrium position and released from rest. Show that in the expression for the displacement $$ x=\mathrm{e}^{-p t}(F \cosh q t+G \sinh q t) $$ where $$ p=\frac{r}{2 m} \quad \text { and } \quad q=\left(\frac{r^{2}}{4 m^{2}}-\frac{s}{m}\right)^{1 / 2} $$ that the ratio $$ \frac{G}{F}=\frac{r}{\left(r^{2}-4 m s\right)^{1 / 2}} $$

Okay, so in this problem we are given the initial displacement of the system and we are required to find the values of two constants which is C1, NC two. Right? So the initial displacement is given by this equation, which is menace. If our menace uh and were doing right. So this is anywhere to em at times C one E power iota omega T. And plus C two. E. Power Menace, Iota omega T. Right? And this is equal to a cars be right When T is equal to zero, right? So when T is equal to zero, this is the initial displacement, Right? And uh C one plus C two is equal to a car spee. Right? So this is the condition and now let the initial velocity of the system to be the derivative. Right? So this is X dirt, which is equal to venice are were to em plus I omega prime, Right? And this is C1 e power E Power Menace are were to m plus iota omega prime, right? Times T. And plus Menace are over to em Menace. Iota omega prime, right, times. This is here too. E power empower. Menace are over to a Menace. Iota omega prime times tree. And this is equal to menace. Omega prime A sign be right. And this is it T equal to Mhm. This is a equal to zero. Right? So we will solve this, right? So this is equal to uh by solving area and rearranging this. This is equal to our war to end eight times cast a time Caspi plus I omega prime, C one Minister too. Right? And this is equal to minus omega prime sign Sign up be right. And uh F R M is very very small, right? Uh our our aim is very very small. Our aim is very very small. Right? And are P is approximately equal to pi by two. So the past time of the above equation is approximately equal to zero. Right? So the past term of the above equation is approximately equal to zero. So this is uh C one manner C two is equal to iota a sign peak. Right? And uh Brandy Bobby Question, C1 and C two are given as palace. Right? So C one is equal to eight times CASS peak plus iota san peak and divided by two. Right? And this is equal to Were to write were two E Power ι P. Right? and C2 is given by eight times Caspi. Menace iota sign P divided by two. And this is equal to a or two E power menace iota P. Right? So this is the value of C one, C two. Sorry, this is C two And this is C1, right? So these are the required to lose right?

In this problem. The differential equation for the rate of change of the with respect to t is proportional to 25 minus t. So I can write DP by DT is equal to K multiplication 25 minus city so I can write d P is equal to K multiplication 25 minus 30 DT going forward and integrating it both side. I can write integration of BP physical too. K Integration 25 bt minus k integration P. D t Not just on integration. I get the value of p H 25 k T minus Katie square by two plus the So the solution for this problem is this 25 KT minus Katie square by two plus C.


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