Having trouble solving this diff eq

[travisr34]

So I am an engineer and its been about two years since I have taken a differential equations class and the method in which to solve this has totally been lost to me. If anyone can help me out that would be great thanks.

m1x1''+k(x2-x1)+b(x2'-x1')+F(t)=0
m2x2''+k(x1-x2)+b(x1'-x2')=0



I need to solve for x1 in terms of all over variables.
Here m1,m2,k,and b are constant. F(t) is some impulse force on the system. This is suppose to model a 2 mass spring damper system with an impulse force on one of the masses.

 

[HallsofIvy]

If you add the two equations, you get m1x1"+ m2x2"+ F(T)= 0. We can interpret that as y"= -F(t) where y= m1x1+ m2x2. You can solve that by integrating F(t) twice.

Once you have found y(t), write y= m1x1+ m2x2 as m1x1= y- m2x2 so that x1= (y- m2x2)m1. Put that into either of the equations and you have a single second order equation for x2. Solve that for x2 and use x1= (y- m2x2)/m1 to find x1.

Notice that in integrating to find y, you will have two "constants of integration". Solving the second order equation for x2 will introduce another two constants.
Solving for x1 does not require any integration and does not introduce new constants. Since the original problem had two second order differential equations, four undetermined constants is exactly what we would expect.