Question

The roots and their multiplicities to the characteristic equation of some homogeneous

linear ODE with constant coefficients are given below. Find the general solution to
the ODE. r1 = −1, k1 = 2r2 = −2, k2 = 3

Answer 1

The general solution to the ODE with the given roots and their multiplicities is y(t) = C1 * e^(-1 * t) + C2 * t^(2-1) * e^(-2 * t).

Step-by-step explanation:

The general solution to the ODE with the given roots and their multiplicities can be found using the formula:

y(t) = C1 * e^(r1 * t) + C2 * t^(k1-1) * e^(r2 * t)

Substituting the given values:

y(t) = C1 * e^(-1 * t) + C2 * t^(2-1) * e^(-2 * t)

where C1 and C2 are arbitrary constants which can be determined by initial conditions or boundary conditions.