Obtain the transfer function of the system if y(t)=e^{-t}-2e^{-2t}+e^{-3t} text{ and } x(t)=e^{-0.5t}

When talking about boundary conditions for partial Differential equations, what does an open boundary mean?

Solve the differential equation ${\displaystyle y^{\prime }=\left|x\right|,y(-1)=2}$

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)${\displaystyle f\left(t\right)=2e^{-7t}\sin \left(5t\right)}$

Find the solution of the given system of differential equations by Operator method or Laplace transform.
${\displaystyle 2\frac{dx}{dt}+\frac{dy}{dt}+x+5y=4t}$
${\displaystyle \frac{dx}{dt}+\frac{dy}{dt}+2x+2y=2}$
${\displaystyle x\left(0\right)=3,y\left(0\right)=-4}$

Find the Laplace transform of ${\displaystyle f\left(t\right)=t{e}^{-t}\sin \left(2t\right)}$
Then you obtain ${\displaystyle F\left(s\right)=\frac{4s+a}{{({(s+1)}^2+4)}^2}}$
Please type in a = ?

I have the following equation
${\displaystyle (x{y}^2+x)dx+(y{x}^2+y)dy=0}$
and I am told it is separable, but not knowing how that is, I went ahead and solved it using the Exact method.
Let ${\displaystyle M=x{y}^2+x\text{and}N=y{x}^2+y}$
${\displaystyle My=2xy\text{and}Nx=2xy}$
${\displaystyle \int M.dx\Rightarrow \int x{y}^2+x=x^2{y}^2+\frac{x^2}{2}+g\left(y\right)}$
Partial of ${\displaystyle (x^2{y}^2+\frac{x^2}{2}+g\left(y\right))\Rightarrow x{y}^2+{g\left(y\right)}^{\prime }}$
${\displaystyle {g\left(y\right)}^{\prime }=y}$
${\displaystyle g\left(y\right)=\frac{y^2}{2}}$
the general solution then is
${\displaystyle C=x^2\frac{y^2}{2}+\frac{x^2}{2}+\frac{y^2}{2}}$
Is this solution the same I would get if I had taken the Separate Equations route?

Solve both
a)using the integral definition , find the convolution
$f\ast g\text{ of }f(t)=\cos 2t,g(t)=e^t$
b) Using above answer , find the Laplace Transform of f*g