Transcription of Rewriting a Second Order Equation as a System of First ...
1 Rewriting a Second Order Equation as a System of First Order equations To rewrite a Second Order Equation as a System of First Order equations , begin with, 0)()()(22=++tkydttdycdttydm or 0)()()(=++tkytyctym&&& and initial conditions 00)(yty=, 00)(vty=& Where x(t) is the vertical displacement of the mass about the equilibrium postion. m is the mass, c is the damping constant, and k is the spring constant. With , we can set 0 mmcC=and mkK=and rewrite the model as: 0)()()(=++tKytyCty&&& Now, we let and )()(1tytz=)()(2tytz&=. Now we can express, )()(21tztz=& and )()()(22tKztCztz =& Putting this together we can write, )(10)(2121tzzCKtzz = && If we let =21)(zztz(t) and =CKA10 we can write the System as, )()(tAztz=& with =00)0(vyz
