Find the laplace transform of the following: Change of Scale text{If } Lleft{f(t)right}=frac{s^2-s+1}{(2s+1)^2(s-2)} text{ , find } Lleft{f(2t)right}

Solve the following differential equations. Use the method of Bernoulli’s Equation. ${\displaystyle x(2x^3+y)dy-6y^{2}dx=0}$

Find f’ in terms of g’.
${\displaystyle f\left(x\right)=x^{2}g\left(x\right)}$

Applications of First Order Differential Equations: A dead body was found within a closed room of a house where the temperature was a constant ${\displaystyle {24}^{\circ }C}$. At the time of discovery, the core temperature of the body was determined to be ${\displaystyle {28}^{\circ }C}$. One hour later, a second measurement showed that the core temperature of the body was ${\displaystyle {26}^{\circ }C}$. The core temperature of the body at the time of death is ${\displaystyle {37}^{\circ }C}$. A. Determine how many hours elapsed before the body was found. B. What is the temperature of the body 5 hours from the death of the victim?

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?

Find the inverse laplace transform of
$\frac{6s+9}{s^2+17}s>0$
$y(t)=\dots$

Determine the values of r or which the given differential equation has solutions of the form ${\displaystyle y=e^{rt},y{}^{″}-3y{}^{″}+2y^{\prime }=0}$

Find the inverse Laplace transform of the given function by using the convolution theorem. $F\left(s\right)=\frac{s}{(s+1)(s^2+4)}$