How to solve this equation y'''-8y"+4y'+48= 4x^2+1 using Laplace Transformation.

Use Laplace transform to solve the following initial value problem:
${\displaystyle y^{″}+5y=1+t,y\left(0\right)=0,y^{\prime }\left(0\right)=4}$
A)${\displaystyle \frac{7}{25}{e}^t\cos \left(2t\right)+\frac{21}{10}{e}^t\sin \left(2t\right)}$
B) ${\displaystyle \frac{7}{2}+\frac{t}{5}-\frac{t}{25}{e}^t\cos \left(2t\right)+\frac{21}{10}{e}^t\sin \left(2t\right)}$
C) ${\displaystyle \frac{7}{25}+\frac{t}{5}}$
D) ${\displaystyle \frac{7}{25}+\frac{t}{5}-\frac{7}{25}{e}^t\cos \left(2t\right)+\frac{21}{10}{e}^t\sin \left(2t\right)}$
E) non of the above
F) ${\displaystyle \frac{7}{25}+\frac{t}{5}-\frac{7}{5}{e}^t\cos \left(2t\right)+\frac{21}{10}{e}^t\sin \left(2t\right)}$

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s. At what rate is his distance from second base decreasing when he is halfway to first base?

Solve the differential equation using either the method of undetermined coefficients or Laplace transforms.
${\displaystyle y-6ytext11y-6y=3x}$

find the Laplace transform of f (t).
${\displaystyle f\left(t\right)=\frac{e^t-e^{-t}}{t}}$

I was toying with equations of the type ${\displaystyle f(x+\alpha )=f^{\prime }\left(x\right)}$ where f is a real function. For example if ${\displaystyle \alpha =\frac{\pi }{2}}$ then the solutions include the function ${\displaystyle f_{\lambda ,\mu }\left(x\right)=\lambda \cos (x+\mu )}$. Are there more solutions?
On the other hand, if I want to solve the equation for any ${\displaystyle \alpha }$, I can assume a solution of the form ${\displaystyle f\left(x\right)=e^{\lambda x}}$, and find ${\displaystyle \lambda }$ as a complex number that enables me to solve the equation...
I was wondering: is the set of the solutions of dimension 2 because the derivative operator creates one dimension and the operator ${\displaystyle \varphi :f\left(x\right)\to f(x+1)}$ adds another one? Is there some litterature about this kind of equations?

A function f (t) is given as. Find the laplace transformation of this function
${\displaystyle f\left(t\right)=e^{-t}\cos 2t\sin t}$

Find the inverse Laplace transform of the following transfer function:
${\displaystyle \frac{Y\left(s\right)}{U\left(s\right)}=\frac{50}{{(s+7)}^2+25}}$
Select one:
a)${\displaystyle f\left(t\right)=10e^{-7t}\sin \left(5t\right)}$
b)${\displaystyle f\left(t\right)=10e^{7t}\sin \left(5t\right)}$
c)${\displaystyle f\left(t\right)=50e^{-7t}\sin \left(5t\right)}$
d)${\displaystyle f\left(t\right)=2e^{-7t}\sin \left(5t\right)}$