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To grasp the concept of a *complementary function*, imagine it as finding the solution to a part of the differential equation without external influence, or more technically, solving the *homogeneous differential equation*. For example, taking the given differential equation: \[x^{\prime \prime}(t) + x(t) = t^{2},\]we focus on the homogeneous part of the equation by ignoring the \(t^2\) term, hence solving:\[x'' + x = 0.\]
This step uses the **auxiliary equation** method. The auxiliary equation here is **\(m^2 + 1 = 0\)**. Solving this quadratic gives the roots \(m = \pm i\). These roots indicate imaginary solutions, leading us to form the complementary function:

• x_c(t) = C_1 \cos(t) + C_2 \sin(t),

where \(C_1\) and \(C_2\) are arbitrary constants.
The complementary function represents all possible solutions of the homogeneous equation and is critical for constructing the general solution.

A *particular integral* is a specific solution to the non-homogeneous differential equation, one that accounts for the non-zero right-hand side in the original equation. Here, we deal with:\[x^{\prime \prime}(t) + x(t) = t^{2}.\]To find a particular integral, we assume a form that matches the right-side polynomial. For this example, we try:\[x = At^2 + Bt + C,\]where \(A\), \(B\), and \(C\) are constants to be determined. Taking derivatives, we have:

• x' = 2At + B
• x'' = 2A

Next, substitute these into the original equation:\[2A + At^2 + Bt + C = t^2.\]From comparing coefficients, we deduce:

• \(A = 1\)
• \(B = 0\)
• \(C = 2\)

Thus, the particular integral becomes:

• x_p(t) = t^2 + 2

The particular integral helps complete the general solution alongside the complementary function.

A *homogeneous differential equation* is one set to zero, focusing purely on the relationship of its own terms without any external addition or forcing term, like the isolated version of:\[x'' + x = 0,\]from our original problem. The homogeneous nature means that:

• All terms involve the sought function \(x(t)\) or its derivatives.
• There are no constant or forcing terms involved (i.e., right side of equation is zero).

To solve a homogeneous differential equation, we employ the **method of auxiliary equation**, which translates the differential equation into an algebraic expression—**\(m^2 + 1 = 0\)** for our example.The solutions to this equation, \(m = \pm i\), are purely imaginary, hinting at oscillating behavior. Consequently, they lead us to represent the solution using:\( x_c(t) = C_1\cos(t) + C_2\sin(t).\)Understanding homogeneous differential equations is crucial as they form the backbone of the general solution, allowing us to superimpose a particular solution to address non-zero forcing terms, ultimately resulting in the complete general solution of the problem.