Anybody know how to solve this?

Question

the differential equation is:
m*x" - c*x' = ps*^(-bt)
where m*x" is acceleration which is d^2x/dt^2 and
-c*x' is velocity which is dx/dt and
ps*^(-bt) = p*s*e^(-bt) which =m*a = net force

Making the following assumptions:
Pressure outside can = p, mass of can = m, Area of can ends (each) = A, area of valve hole = s

Answer 1

Taking
mx´´-cx´=ps*e^-bt
The caracteristic equation is
mr^2-cr=0 so r= 0 and r= c/m
The solution of the homogeneous equation (without second side is
x= C_1+C_2 *e^(c/m)t Let´s find a particular solution of the equation with right side
It is logic to look for x= ke^-bt

x´= -bke^-bt
x´´=b^2ke^-bt
so mb^2*k +cbk=ps so k =ps/(mb^2+cb) and the complete solutionis
x = C_1+C_2*e^(c/m*t)+ps/(mb^2+cb) *e^-bt
C_1 and C_2 are constants depending on initial conditions