Final answer:
The solutions to 3.1 homogeneous differential equations with constant coefficients are found by first obtaining the characteristic equation using the auxiliary equation. The solutions are then given by a linear combination of exponential functions involving the roots of the characteristic equation.
Step-by-step explanation:
In the context of 3.1 homogeneous differential equations with constant coefficients, the solutions can be found by first finding the characteristic equation using the auxiliary equation. The characteristic equation is obtained by substituting y(x) = erx into the differential equation and solving for r. The solutions to the homogeneous differential equation are then given by y(x) = C1er1x + C2er2x + ... + Cnernx, where r1, r2, ..., rn are the roots of the characteristic equation and C1, C2, ..., Cn are arbitrary constants.