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Solve the initial value problem $$\style{font-size:18pt}{\displaystyle \frac{dy}{dx} + (1+y^2)\, e^{4x} =0}$$ with $$\style{font-size:18pt}{y(0) = 0.}$$

Use the definition of the Laplace transform to calculate $$\style{font-size:18pt}{{\cal L}\left[e^{3t}\right].}$$For what values of $$\style{font-size:18pt}{s}$$ does this transform exist? Justify your answer based on your calculations. 

In People v. Lauria, the Court held that:

A student finds a news article about a recent event that relates to their discussion on Jonathan Swift. The student copies the quote directly out of the news article but doesn’t use quotations or a parentheses tag.

A student finds a news article about a recent event that relates to their discussion on Jonathan Swift. The student introduces a quote from the news article with “According to WKRG.”

Use any method we’ve covered to solve the initial value problem$$\style{font-size:18pt}{y” + 4y’+ 7y = 0,}$$with $$\style{font-size:18pt}{y(0)=2}$$ and $$\style{font-size:18pt}{y'(0)=0.}$$

Find the general solution of the differential equation $$\style{font-size:18pt}{\displaystyle x\, \frac{dy}{dx} + 2 y = 4x^2.}$$Assume that $$\style{font-size:18pt}{x\ge 0.}$$

Let $$\style{font-size:18pt}{f(t)=\sin(at)}$$ and $$\style{font-size:18pt}{g(t)=1,}$$ where $$\style{font-size:18pt}{a}$$ is a positive constant. (a) Use the definition of the convolution to show that $$\style{font-size:18pt}{(f\ast g)(t) = \dfrac{1}{a}\left( 1- \cos(at) \right).}$$(b) Use the result from (a) to find $$\style{font-size:18pt}{{\cal L}^{-1}\left[\dfrac{3}{s(s^2+2)}\right].}$$

Dementia is a disease that is defined in the following way:

According to the film, John Q, how much money would it cost to perform Mike’s procedure?