Classify the differential equation. Determine the order,...

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or nonhomogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$y^{\prime \prime}-3 y+2 y=\cos (t)$$

Key Concepts

Order of a Differential Equation

The order of a differential equation is determined by the highest derivative of the unknown function that appears in the equation. Knowing the order is fundamental because it gives insight into the behavior of the solutions and guides the selection of appropriate solution methods.

Linearity of a Differential Equation

A linear differential equation is one in which the unknown function and all its derivatives appear to the first power, and there are no products or nonlinear functions (like exponents or trigonometric functions) of the unknown function or its derivatives. This property is significant because linear equations have superposition principles which allow the construction of general solutions from simpler ones.

Homogeneous vs. Nonhomogeneous Differential Equations

A homogeneous differential equation is one in which every term involving the unknown function and its derivatives sums to zero, implying that the zero function is a solution. Conversely, a nonhomogeneous differential equation has additional terms (called forcing functions or input functions) that do not depend on the unknown function. This distinction is crucial since it influences the structure of the solution, often requiring the sum of the complementary (homogeneous) solution and a particular solution.

Characteristic Equation

For second-order linear homogeneous differential equations with constant coefficients, the characteristic equation is a polynomial equation obtained by substituting an exponential trial solution into the differential equation. It is vital because its roots determine the form of the general solution, which may include real, repeated, or complex conjugate roots.

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