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?

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

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)

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)

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)

Partial differential equation

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

Partial differential equation

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

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)

Partial differential equation

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

Partial differential equation

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