What is the general solution to the differential equation describing a mass-spring-damper? t=time x= extension of spring M=Mass K=Spring Constant C=Damping Constant g= acceleration due to gravity Spring has 0 length under 0 tension Spring has 0 extension at t = 0 If the Force downwards due to the mass equals: Force downwards = M*g And this is opposed by tension in the spring: Force upwards = - K*x And there is a 3rd force acting on the system caused by the damper. This allways acts in the opposite direction as velocity. Force = - C*(dx/dt) //because damping is proportional to velocity. So the overall force acting on the mass equals: Net Force = M*g - K*x - C*(dx/dt) This net force causes an acceleration in the direction of the force: F=Ma so... M*(d2x/dt2) = M*g - K*x - C*(dx/dt) \\you can see that this will oscillate, because when x is 0 the acceleration will be positive, when M*g= K*x + C*(dx/dt), the acceleration will be 0, and then become negative, and then the velocity will become 0, and then negative, and then the x will become 0 again, and it will repeat. So the differential equation describing this system is: M*(d2x/dt2) + C*(dx/dt) + K*x = Mg Mx''+Cx'+Kx=Mg This is a second order linear non homogeneous differential equation. Mx''+Cx'+Kx=g(t) where g(t) is a constant. Now my question is, how do I solve it for x? I tried using the undetermined coefficients method, but I do not get an equation that implies that x is oscillating. Shouldn't I get a sin or cos somewhere in the answer? Here is what I tried: corresponding Homogeneous Equation: Mx''+Cx'+Kx=0 characteristic equation = M*(r^2) + C*r + K General solution to the homogeneous equation: C1(e^pt)+C2(e^qt) where p=(-C+SQRT((C^2)-4*M*K))/2M q=(-C-SQRT((C^2)-4*M*K))/2M //from the quadratic equation Now to calculate the particular solution: Mx''+Cx'+Kx=Mg The right hand side is a polynomial of degree 0. try a polynomial as the solution x=Ax^2 + Bx + C x'=2Ax +B x''=2A Sub these in to the differential equation and you end up with: x=Mg/K So the general solution to the differential equation is: x(t) = C1(e^pt)+C2(e^qt) + Mg/K This can't be correct, because x should oscillate over time. C1, C2, p, q, ang Mg/K are all constants so this equation does not describe an oscillation. Where did I go wrong? Thanks!