second order, constant coeff. homogeneous, one root repeated
Let and plug into the above and simplify, we obtain the charaterstic equation
Repated root. Hence the two L.I. basis solutions are
And the homogeneous solution is
second order, constant coeff. homogeneous, two real distinct roots
Let and plug into the above and simplify, we obtain the charaterstic equation
Hence the L.I. basis solutions are
And the homogeneous solution is
second order, constant coeff. homogeneous, two complex conjugate roots
Let and plug into the above and simplify, we obtain the charaterstic equation
Whose roots are
Hence the L.I. basis solutions are
the homogeneous solution is
This can be converted to real basis using Euler relation which results in