Identify each of the differential equations as to type (for example, separable, linear first order, linear second order, etc.), and then solve it. \(y^{\prime \prime}-2 y^{\prime}+5 y=5 x+4 e^{x}(I+\sin 2 x)\)

In mathematics, differential equations are used to describe various physical phenomena. A linear differential equation is one where the unknown function and its derivatives appear to the power of one (hence, ‘linear’). These equations can be first-order or second-order based on the highest derivative's order. For example, \(y'' - 2y' + 5y = 5x + 4e^x(I + \sin 2x)\) is a second-order linear differential equation because the highest derivative is the second derivative, and all coefficients of \(y\) and its derivatives are constants.
A distinguishing feature of linear differential equations is their applicability in systems that can be expressed linearly. This means the superposition principle holds, where the sum of solutions is also a solution.
Linear differential equations can be categorized into homogeneous and non-homogeneous. The given example is a non-homogeneous linear differential equation since the right-hand side is not zero. The general solution of such an equation comprises two parts: the complementary solution that addresses the homogeneous part, and the particular solution that satisfies the non-homogeneous part.

The complementary solution (\(y_c\)) of a differential equation handles the homogeneous part (i.e., the part where the right-hand side is zero). For the given equation, this is expressed as \(y'' - 2y' + 5y = 0\).
To find the complementary solution, we solve the characteristic equation. Here, we get the characteristic equation by substituting \(y = e^{rt}\), leading to \(r^2 - 2r + 5 = 0\).
Solving this quadratic equation, we get:

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