## Mathematica - Analysis of a mass-damper-spring system

Hello,

I am currently working on a problem, but at the moment I am stuck. I just don't know how to solve the problem so I hope someone can help me with it. This is the question

1a : Consider the standard mass-damper-spring system:
m y''+γ y'+k y=u

where the constants have the following values:
m=28
γ=3
k=13

First find the homogeneous solution to the differential equation using the DSolve command. This is also known as the transient solution. In other words, find the solution where u(t)=0. Use the initial values y(0)=1 and y^\[Prime](0)=1.

(* Use Set to assign your answer to the variable before this comment *)(* \
Your answer should be in the default output form produced by DSolve, \
which is a Rule within Lists.
The form should look similar to the following: *)

1b : What is the natural frequency Subscript[\[Omega], 0] of this system?

I hope someone can help me

Greets
 This is what I have so far DSolve[28 y''[t] + 3 y'[t] + 13 y[t] == 0, y[t], t] {{y[t] -> E^(-3 t/56) C[2] Cos[(Sqrt[1447] t)/56] + E^(-3 t/56) C[1] Sin[(Sqrt[1447] t)/56]}} Now is my question, how do I lose the constants C[2] and C[1]. I think it has something to do with that y[0]=1 and y'[0]=1, but I don't know how I should insert those values in the equeation.
 Mentor You can do it one of two ways. The first is to take your output function and use it to evaluate y[0]==1 and y'[0]==1. That is two equations which you can solve for C[1] and C[2]. The other way is to add those initial conditions to the equations in DSolve. So instead of just solving the differential equation, make a list like {diffeq, y[0]==1, y'[0]==1} or {diffeq, y[0]==y0, y'[0]==v0}

## Mathematica - Analysis of a mass-damper-spring system

Ok, but how do I exactly add those initial conditions in the equation

I have now : DSolve[28''y[t]+3y'[t]+13y[t]==0,y[t],t]

So where in the equation has the ''y[0]==1 andd y'[0]==1 be placed and which brackets do I have to use???
 Oh I think I got it correct now : DSolve[{28 y''[t] + 3 y'[t] + 13 y[t] == 0, y[0] == 1, y'[0] == 1}, y[t], t] and now there no constants anymore in the equation. But now is the next question : What is the natural frequency w0 of this system? Someone who knows that?
 Your mass is 28; spring constant or stiffness is 13. Radian natural frequency is sqrt(k/m) = 0.681 radians/sec or 0.1084 Hertz (cycles/sec).

Recognitions: