91Ó°ÊÓ

A second-order linear differential equation involves the second derivative of a function as the highest-order derivative. The equation can be expressed in the standard form \[ \frac{d^2y}{dx^2} + p(x) \frac{dy}{dx} + q(x)y = f(x) \] where \(y\) is the dependent variable, and \(f(x)\) is a known function which makes the equation non-homogeneous if it's not equal to zero.
In our original problem, the equation is \[ \frac{d^2y}{dx^2} + y = \sin x \] This is a simple second-order differential equation without a first derivative term. It is non-homogeneous because of the \(\sin x\) term.
Understanding the behavior of such equations is crucial as they often model real-world phenomena like oscillations and electrical circuits.

The method of undetermined coefficients is a technique used to find a particular solution of non-homogeneous linear differential equations. This method is especially useful when the non-homogeneous part, \(f(x)\), is a simple function such as exponential, sine, cosine, or polynomial functions.Here's how it works:

• Assume a specific form of the solution based on the form of \(f(x)\).
• Substitute this assumed solution into the differential equation.
• Adjust the coefficients in your assumed solution until both sides of the equation match.

In our case, we guessed a particular solution for \(\sin x\) and adjusted the coefficient to find that \(C = -\frac{1}{2}\). This led to the particular solution \[ y_p(x) = -\frac{1}{2}x\cos x \]

The complementary function is part of the general solution of a non-homogeneous differential equation. It represents the solution of the associated homogeneous equation obtained by setting the non-homogeneous part \(f(x)\) to zero.To find it:

• Solve the homogeneous equation: \( \frac{d^2y}{dx^2} + y = 0 \).
• Form the auxiliary equation from the differential operator terms: \(m^2 + 1 = 0 \).
• Find the roots of the auxiliary equation. Here, they are complex: \(m = \pm i\).
• Thus, the complementary function is \(y_c(x) = A \sin x + B \cos x\), where \(A\) and \(B\) are constants determined by initial conditions.

The auxiliary equation plays a key role in solving linear differential equations. It's derived from the homogeneous differential equation by replacing the derivative terms with powers of a variable, usually \(m\).For equations of the form:\[ \frac{d^2y}{dx^2} + ay = 0 \] The corresponding auxiliary equation is obtained by substituting \(y = e^{mx}\), leading to \[ m^2 + a = 0 \] In our case:

• We have \( \frac{d^2y}{dx^2} + y = 0 \).
• This leads to the auxiliary equation \(m^2 + 1 = 0\).
• Solving gives roots \(m = \pm i\), indicating oscillatory solutions.
• The complex roots result in the complementary function involving sine and cosine terms.

Understanding the auxiliary equation helps predict the nature of solutions for the homogeneous part of the differential equation.