Aneeka Hunt
2021-01-16
Answered

Solve differential equation
$\frac{dy}{dx}-\left(\mathrm{cot}x\right)y={\mathrm{sin}}^{3}x$

You can still ask an expert for help

AGRFTr

Answered 2021-01-17
Author has **95** answers

Jeffrey Jordon

Answered 2021-12-12
Author has **2262** answers

Answer is given below (on video)

Jeffrey Jordon

Answered 2021-12-14
Author has **2262** answers

Answer is given below (on video)

Jeffrey Jordon

Answered 2021-12-14
Author has **2262** answers

Answer is given below (on video)

asked 2021-02-24

What would be the laplace transform of a function of f or $L\{f(t)\}$ ?

asked 2021-09-18

to find the inverse Laplace transform of the given function.

$F\left(s\right)=\frac{{2}^{n+1}n!}{{s}^{n+1}}$

asked 2020-11-29

Part II

29.[Poles] (a) For each of the pole diagrams below:

(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.

(ii) Write down two examples of such f(t) and F(s).

The diagrams are:$(1)1,i,-i.(2)-1+4i,-1-4i.(3)-1.(4)$ The empty
diagram.

(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by$w(t)=u(t){e}^{-\frac{t}{2}}\mathrm{sin}(\frac{3t}{2})$

(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?

(ii) Sketch the pole diagram of the system.

(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?

29.[Poles] (a) For each of the pole diagrams below:

(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.

(ii) Write down two examples of such f(t) and F(s).

The diagrams are:

(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by

(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?

(ii) Sketch the pole diagram of the system.

(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?

asked 2022-05-17

We have ${y}^{\prime}=x/y$, which is a first-order homogeneous differential equation.

It can be solved by rearranging to y dy=x dx and then integrating both parts which yields that $y=\pm \sqrt{{x}^{2}+c}$.

Now if we use the substitution $y=ux\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{y}^{\prime}={u}^{\prime}x+u,$, and rewrite the differential equation as

${u}^{\prime}x+u=\frac{1}{u}$

and then rearrange to

$\left(\frac{1}{1/u-u}\right)du=\left(\frac{1}{x}\right)dx$

by integrating both parts we get that

$\begin{array}{}\text{(1)}& -\frac{1}{2}\mathrm{ln}|{u}^{2}-1|=\mathrm{ln}\left|x\right|+c\end{array}$

For $y=\pm x$ (a special solution for c=0) $\to u=\pm 1$, and by plugging $\pm 1$ into (1) we get that

$\begin{array}{}\text{(2)}& \mathrm{ln}\left|0\right|=\mathrm{ln}\left|x\right|+c\end{array}$

What does equation (2) mean? $\mathrm{ln}\left|0\right|$ is undefined. Is this of any significance?

Edit 1:

As pointed out when rearranging from ${u}^{\prime}x+u=\frac{1}{u}$ to $\left(\frac{1}{1/u-u}\right)du=\left(\frac{1}{x}\right)dx$, we implicitly assumed that $u\ne \pm 1$. Equation (1) does not hold for $u=\pm 1$

Edit 2:

Solving equation (1) for u with $u\ne \pm 1$, we arrive at the same family of equations but with $c\ne 0$. The fact that c can be zero comes from setting $u=\pm 1$

It can be solved by rearranging to y dy=x dx and then integrating both parts which yields that $y=\pm \sqrt{{x}^{2}+c}$.

Now if we use the substitution $y=ux\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{y}^{\prime}={u}^{\prime}x+u,$, and rewrite the differential equation as

${u}^{\prime}x+u=\frac{1}{u}$

and then rearrange to

$\left(\frac{1}{1/u-u}\right)du=\left(\frac{1}{x}\right)dx$

by integrating both parts we get that

$\begin{array}{}\text{(1)}& -\frac{1}{2}\mathrm{ln}|{u}^{2}-1|=\mathrm{ln}\left|x\right|+c\end{array}$

For $y=\pm x$ (a special solution for c=0) $\to u=\pm 1$, and by plugging $\pm 1$ into (1) we get that

$\begin{array}{}\text{(2)}& \mathrm{ln}\left|0\right|=\mathrm{ln}\left|x\right|+c\end{array}$

What does equation (2) mean? $\mathrm{ln}\left|0\right|$ is undefined. Is this of any significance?

Edit 1:

As pointed out when rearranging from ${u}^{\prime}x+u=\frac{1}{u}$ to $\left(\frac{1}{1/u-u}\right)du=\left(\frac{1}{x}\right)dx$, we implicitly assumed that $u\ne \pm 1$. Equation (1) does not hold for $u=\pm 1$

Edit 2:

Solving equation (1) for u with $u\ne \pm 1$, we arrive at the same family of equations but with $c\ne 0$. The fact that c can be zero comes from setting $u=\pm 1$

asked 2021-01-27

Cosider the system of differential equations

Convert this system to a second order differential equations and solve this second order differential equations

asked 2022-01-19

Dont

asked 2022-01-16

Solve the Homogenous Differential Equations.

$(x-y\mathrm{ln}y+y\mathrm{ln}x)dx+x(\mathrm{ln}y-\mathrm{ln}x)dy=0$