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2021-02-16
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Laaibah Pitt

Answered 2021-02-17
Author has **98** answers

asked 2022-01-22

Find the general solution to these first order differential equations.

$3(3{x}^{2}+{\u0443}^{2})dx-2xydy=0$

asked 2021-11-15

Differentiate.

$y=\mathrm{sec}\left(\theta \right)\mathrm{tan}\left(\theta \right)$

asked 2022-03-23

Consider the following second-order linear homogeneous difference equation with constant coefficients and two initial conditions:

${u}_{n+2}-6{u}_{n+1}+9{u}_{n}=0,\text{}{u}_{0}=1,\text{}{u}_{1}=9$

Determine the sequence$u}_{2},\text{}{u}_{3},\text{}{u}_{4},\text{}{u}_{5$ . Solve the difference equation for $u}_{n$ and use the result to check for $u}_{5$

Determine the sequence

asked 2020-12-29

Find ${L}^{-1}\left\{\frac{1}{(s\text{}+\text{}6)(s\text{}-\text{}4)}\right\}$

asked 2021-12-31

Identify the characteristic equation, solve for the characteristic roots, and solve the 2nd order differential equations.

$(18{D}^{3}-33{D}^{2}+20D-4)y=0$

asked 2022-07-01

I am trying to solve a first order differential equation with the condition that $g(y)=0$ if $y=0$:

$\begin{array}{rl}& a{g}^{\prime}(cy)+b{g}^{\prime}(ey)=\alpha \\ \text{(1)}& & g(0)=0,\end{array}$

where parameters a,b,c,e are real nonzero constants; $\alpha $ is a complex constant; function $g(y):\mathbb{R}\to \mathbb{C}$ is a function mapping from real number y to a complex number. The goal is to solve for function $g(\cdot )$. This is what I have done. Solve this differential equation by integrating with respect to y:

$\begin{array}{r}\frac{a}{c}g(cy)+\frac{b}{e}g(ey)=\alpha y+\beta ,\end{array}$

where $\beta $ is another complex constant. Plugging in y=0 and using the fact that g(0)=0, we have $\beta =0$. Therefore, we have

$\begin{array}{r}\frac{a}{c}g(cy)+\frac{b}{e}g(ey)=\alpha y.\end{array}$

The background of this problem is Cauchy functional equation, so my conjecture is one solution could be $g(y)=\gamma y$. Plugging in $g(y)=\gamma y$, I get $\gamma =\frac{\alpha}{a+b}$, which implies that one solution is $g(y)=\frac{\alpha}{a+b}y$. Then, I move on to show uniqueness. I define a vector-valued function $h\equiv ({h}_{1},{h}_{2}{)}^{T}$ such that

$\begin{array}{rl}{h}_{1}(y)& =\frac{a}{c}{g}_{1}(cy)+\frac{b}{e}{g}_{1}(ey)\\ {h}_{2}(y)& =\frac{a}{c}{g}_{2}(cy)+\frac{b}{e}{g}_{2}(ey),\end{array}$

where $g(y)\equiv {g}_{1}(y)+i{g}_{2}(y)$. Then, I rewrite this differential equation as

$\begin{array}{rl}& {h}^{\prime}(y)=\alpha \\ \text{(2)}& & h(0)=0,\end{array}$

where $\alpha \equiv ({\alpha}_{1},i{\alpha}_{2}{)}^{T}$. By the uniqueness theorem of first order differential equation, solution h(y) is unique. I have two questions. First, I think equation (1) and (2) should be equivalent. However, it seems that equation (1) can imply equation (2) but equation (2) may not imply equation (1). This is because h(0)=0 may imply either ${g}_{1}(0)=0,{g}_{2}(0)=0$ or $\frac{a}{c}+\frac{b}{e}=0$. Second, I have only proved that h(y) is unique. How should I proceed to show g(y) is also unique.

$\begin{array}{rl}& a{g}^{\prime}(cy)+b{g}^{\prime}(ey)=\alpha \\ \text{(1)}& & g(0)=0,\end{array}$

where parameters a,b,c,e are real nonzero constants; $\alpha $ is a complex constant; function $g(y):\mathbb{R}\to \mathbb{C}$ is a function mapping from real number y to a complex number. The goal is to solve for function $g(\cdot )$. This is what I have done. Solve this differential equation by integrating with respect to y:

$\begin{array}{r}\frac{a}{c}g(cy)+\frac{b}{e}g(ey)=\alpha y+\beta ,\end{array}$

where $\beta $ is another complex constant. Plugging in y=0 and using the fact that g(0)=0, we have $\beta =0$. Therefore, we have

$\begin{array}{r}\frac{a}{c}g(cy)+\frac{b}{e}g(ey)=\alpha y.\end{array}$

The background of this problem is Cauchy functional equation, so my conjecture is one solution could be $g(y)=\gamma y$. Plugging in $g(y)=\gamma y$, I get $\gamma =\frac{\alpha}{a+b}$, which implies that one solution is $g(y)=\frac{\alpha}{a+b}y$. Then, I move on to show uniqueness. I define a vector-valued function $h\equiv ({h}_{1},{h}_{2}{)}^{T}$ such that

$\begin{array}{rl}{h}_{1}(y)& =\frac{a}{c}{g}_{1}(cy)+\frac{b}{e}{g}_{1}(ey)\\ {h}_{2}(y)& =\frac{a}{c}{g}_{2}(cy)+\frac{b}{e}{g}_{2}(ey),\end{array}$

where $g(y)\equiv {g}_{1}(y)+i{g}_{2}(y)$. Then, I rewrite this differential equation as

$\begin{array}{rl}& {h}^{\prime}(y)=\alpha \\ \text{(2)}& & h(0)=0,\end{array}$

where $\alpha \equiv ({\alpha}_{1},i{\alpha}_{2}{)}^{T}$. By the uniqueness theorem of first order differential equation, solution h(y) is unique. I have two questions. First, I think equation (1) and (2) should be equivalent. However, it seems that equation (1) can imply equation (2) but equation (2) may not imply equation (1). This is because h(0)=0 may imply either ${g}_{1}(0)=0,{g}_{2}(0)=0$ or $\frac{a}{c}+\frac{b}{e}=0$. Second, I have only proved that h(y) is unique. How should I proceed to show g(y) is also unique.

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?