posted 1 years ago

\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

posted 1 years ago

\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)

posted 1 years ago

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

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

posted 1 years ago

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)

posted 1 years ago

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?

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posted 1 years ago