Final answer:
For second-order differential equations, solutions are found by forming a characteristic equation based on the given equation. The discriminant of this characteristic equation determines the form of the general solution. Specific initial conditions and constants are necessary to finalize the solution to a particular equation.
Step-by-step explanation:
Second-order differential equations of the form y'' + ay' + by = 0, y'' - ay' + by = 0, y'' + ay' - by = 0, and y'' - ay' - by = 0 can be solved using the characteristic equation method. The characteristic equation is a quadratic equation derived from substituting a trial solution of the form y = ert into the differential equation, where r is a constant to be determined.
For each case, the characteristic equation is as follows:
- For (a) r2 + ar + b = 0
- For (b) r2 - ar + b = 0
- For (c) r2 + ar - b = 0
- For (d) r2 - ar - b = 0
These characteristic equations can be solved using the quadratic formula. Depending on the discriminant, which is a2 - 4b, the general solution will vary. If the discriminant is positive, there will be two real and distinct roots, leading to a solution of the form y = C1er1t + C2er2t. If the discriminant is zero, there will be a repeated real root, and the solution will be y = (C1 + C2t)ert. If the discriminant is negative, the roots will be complex, resulting in a solution involving sines and cosines.
To determine which option from (a), (b), (c), or (d) is the correct one, additional initial conditions and specific values of the constants a and b would be required. In the absence of such specifics, we can only provide the general procedure for finding the solutions.