Find Homogeneous Differential Equation From Two Dynamic Equations

Question

Equation #1 $$ y^{''}+C_1t^{''}+C_2y = C_3y^{'}+C_4t $$ Equation #2 $$ C_5y^{''}+C_6t^{''}+C_7t = C_8y^{'}+C_9t $$ Both y and t are dynamic, and I need to somehow get these two equations into homogeneous differential equations. It has been a while since I've taken Diff Eq, so I might be missing something obvious.I can't seem to remember how to do differential equations with 2 dynamic variables so I dont even know how to start.

Thanks

Answer 1

You already have a homogeneous equation/system. It is, in matrix-vector form, of the type $$ Mu''+Nu'+Ru=0 $$ $u=(y,t)^T$, and has to be transformed as $$ u''+M^{-1}Nu'+M^{-1}Ru=0 $$ to get the system into explicit form.