Final answer:
To solve a homogeneous differential equation, one utilizes methods aligned with the form of the equation, which might involve exponential, trigonometric, or logistic functions. The solution's correctness is verified by substituting derivatives back into the original equation.
Step-by-step explanation:
To find the general solution to a homogeneous differential equation, you would typically use a method appropriate for the type of equation you are dealing with. Common forms for solutions include exponential functions for first-order linear homogeneous differential equations and trigonometric functions for second-order linear homogeneous differential equations with constant coefficients. In some cases, such as for the logistic differential equation mentioned, the solution takes the specific form of a logistic curve.
When tackling specific solution forms such as the quadratic equations, solutions often take the form of a second-order polynomial or quadratic function. To prove a solution's validity, you would usually derive the first and second derivatives, and substitute them back into the original equation to verify whether they satisfy the equation.
Overall, the process involves stating the formula or equation, finding the derivatives if necessary, and then substituting known values to solve for the unknowns, whether that be concentrations, variables, or constants like h.