Verified

Partial differential equation

Consider the following ODE with given IC:

Y^{\prime}(x)=x^{2} \cos (Y(x))^{2}, Y(0)=1

\text { and answer the following questions: }

\text { What is } \frac{\partial f(x, z)}{\partial z} ?

c) Find the analytical solution Y(x) and verify where it exists.

In what region of x will the solution exist?

Verified

Partial differential equation

4. Find the solution and final value (when t→ 0) of the following differential equation by using Laplace transform. Include the mathematical method to justify your solution and state any properties of Laplace Transforms that you have used to perform your manipulations. (12)

\frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+c x=2 e^{-4 t}

\text { Where } \frac{d x}{d t}=1 \text { and } x=0 \text { when } t=0

b=3 \text { and } c=3

Verified

Partial differential equation

3. The vibration of a cable supporting a suspension bridge can be described by the one- dimensional wave equation,

\frac{\partial^{2} u}{\partial t^{2}}=\alpha^{2} \frac{\partial^{2} u}{\partial x^{2}}

The problem has the following boundary and initial conditions:

• Is the trial solution u(x,t) = g(t)sin(-x) sensible for this problem, discuss10why/why not. (3)

\text { Using the trial solution } u(x, t)=g(t) \sin \left(\frac{n \pi}{10} x\right), \text { convert the wave equation }

into a single ODE and find its general solution. (6)

c) Write the general solution to the PDE and solve for the unknown constants. (6)

Verified

Partial differential equation

2. A support for electrified railway cables is cantilevered from the side of the track by a beam with spring stiffness k. The mass of the beam is 3M and is assumed to be concentrated at the free end. The cable, of mass 2M, is supported by a spring of

stiffness k from the end of the cantilever.

The system of equations governing the motion of the system is:

3 M y_{1}^{\prime \prime}=-2 k y_{1}+k y_{2}

2 M \ddot{y}_{2}=k y_{1}-k y_{2}

k = 22

Write the above system of differential equations in matrix form. Then, by considering the trial solution: y = e"X, show

that system can be written as an eigenvalue problem. (3)

b) Find the general solution for the system of equations by solving the eigenvalue problem. (12)

Verified

Partial differential equation

\text { Consider the piecewise periodic function } f(t) \text { with a period of } 2 \pi \text { : }

a) Write the Fourier series expansion of f(t).

c) Write the Fourier series expansion found in (a) in Amplitude-phaseform. (3)

Verified

Partial differential equation

2d. Explain why this function is a one-to-one function (10 pts)

Verified

Partial differential equation

2b Demonstrate how this function works using the domain, and range. (5 pts)

Verified

Partial differential equation

2a. Give your own real life example of a one-to-one function. State what the domain and range values represents. (10 pts)

Verified

Partial differential equation

1c. Explain why it is necessary for this function to be defined piece-wise (6 pts)

Verified

Partial differential equation

1b. Calculate f(x) for two values of x. Explain what these results mean. (12 pts)

**Kindly submit your queries**

we will make sure available to you as soon as possible.

Search Other Question

Getting answers to your urgent problems is simple. Submit your query in the given box and get answers Instantly.

Success

- Offers
- Flash sale on now! Get
**20%**off until**25th June**, online at TutorBin. Use discount code**ALK&8JH**at**Tutorbin.com/Booking** - News
- Latest Blog Published:

[Blog Name], online at [Time] - News
- Latest Blog Published:

[Blog Name], online at [Time]