Question

Select a topic from a list of six possible topics on British literature. Your topic will connect the work of literature to the time and culture in which it was written. Decide on a research question, develop a thesis, and conduct your research. Cite at least four research sources. At least one of them must be a print source that is not an encyclopedia. Develop an organizational plan, including a formal outline for your paper. Write a research paper on an aspect of British literature that you choose. It will be 6–9 double-spaced pages long, or about 1,800–2,700 words.

Answer 1

The solution to the given initial value problem is
`\( y(t) = e^t(A\cos t + B\sin t) + (1)/(2) \cos t \), where \( A \) and \( B \)` are constants determined by the initial conditions.

Step-by-step explanation:

In solving the given initial value problem
`\( y'' - 2y' + 2y = \cos t \)`, we can first find the characteristic equation by substituting
`\( y = e^(rt) \)` into the homogeneous part of the equation, leading to
`\( r^2 - 2r + 2 = 0 \)`. Solving this quadratic equation yields complex roots
`\( r = 1 \pm i \)`. The general solution for the homogeneous equation is then
`\( y_h(t) = e^t(A\cos t + B\sin t) \),` where
`\( A \) and \( B \)` are constants.

To find a particular solution for the inhomogeneous part
`\( \cos t \)`, we can assume
`\( y_p(t) = C\cos t + D\sin t \)`. By substituting this into the original differential equation, we find
`\( C = (1)/(2) \) and \( D = 0 \)`. Therefore, the particular solution is
`\( y_p(t) = (1)/(2) \cos t \)`.

The general solution is the sum of the homogeneous and particular solutions,
`\( y(t) = y_h(t) + y_p(t) \)`. Applying the initial conditions will determine the values of the constants
`\( A \) and \( B \)`, resulting in the final solution
`\( y(t) = e^t(A\cos t + B\sin t) + (1)/(2) \cos t \).`