Converting a Higher Order ODE Into a System of First Order ODEs

This paragraph introduces the process of transforming a fourth-order linear constant-coefficient differential equation into a system of first-order differential equations. The initial step involves isolating the highest order derivative on one side of the equation, which in this case is the fourth derivative. The equation is then rewritten with all terms involving derivatives of varying orders moved to the opposite side, effectively reducing their order. The next step is to rename each derivative of the original function, denoted as X, as new variables (x1 through x4), representing the function itself and its first, second, and third derivatives, respectively. The fourth derivative is then equated to the derivative of the third renamed variable (x4 prime). The paragraph concludes with the observation that the original fourth-order equation can now be expressed as a system of first-order equations in terms of these new variables.

The second paragraph delves deeper into the conversion process by illustrating how to derive a system of first-order differential equations from the fourth-order equation. It begins by defining the relationship between the renamed variables and their derivatives, such as x1 prime being equal to x2, and so on. This leads to the identification of three additional first-order differential equations involving the new variables x1, x2, x3, and x4. The paragraph then demonstrates how to represent these relationships in matrix form, with the left side of the matrix equation representing the vector of the variables' derivatives (X prime) and the right side being the product of a matrix (A) and the vector of the variables (X). The matrix A is constructed based on the coefficients of the derivatives in the original equation. The paragraph concludes with the representation of the system of equations in a compact matrix form, showcasing the transformation of the original higher-order differential equation into a manageable system of first-order equations.

The final paragraph extends the discussion to include initial conditions in the system of differential equations derived from a second-order equation, which is an initial value problem. The process starts by isolating the highest derivative, in this case, the second derivative (x2 prime), and rearranging the terms to express it in terms of the original function and its first derivative. The variables are then renamed as before, with x1 representing the function and x2 representing its first derivative. The paragraph highlights the need to correct a mistake in the transcription of the equation and proceeds to establish the system of differential equations with the inclusion of a non-homogeneous term, cosine 3t. The system is then represented in vector and matrix form, with the addition of a forcing function vector. The initial conditions are carefully incorporated, with the initial values for x1 and x2 being specified. The paragraph concludes by presenting the entire initial value problem in vector form, emphasizing the transition from the original equation to a system of equations that can be solved with given initial conditions.