I want to find the laplace inverse of

Elementary Applications of Differential Equations 1. A culture of bacteria consists of 1000 bacteria. After 2 hrs., the culture becomes 300 bacteria. a. How many bacteria remain after 4 hours? b. In how many hours will the culture become 50 bacteria?

Use laplace transform to solve the given system
$a)\frac{dx}{dt}-2x-\frac{dx}{dt}-y=6e^{3t}$
$2\left(\frac{dx}{dt}\right)-3x+\frac{dx}{dt}-3y=6e^{3t}x(0)=3y(0)=0$

Solve for the following problems using substitution suggested by the equation. ${\displaystyle (3x-2y+1)dx+(3x-2y+3)dy=0}$

Solve the third-order initial value problem below using the method of Laplace transforms
$y^{‴}+3y^{″}-18y^{\prime }-40y=-120$
$y(0)=6$
$y^{\prime }(0)=45$
$y"(0)=-21$

Find the general solution of the given differential equation.
${\displaystyle x^2{y}^{\prime }+x(x+7)y=e^x}$
${\displaystyle y=}$

Obtain the Differential Equation from the family of ellipses centered at (h,k) whose semi-major axis is twice the semi-minor axis.