91Ó°ÊÓ

The differential operator is a key concept in solving differential equations. In this context, it is denoted by the symbol \(D\) and represents differentiation with respect to \(x\). When you see a term like \(D^2\), it means taking the second derivative of a function. Here's a simple breakdown:

• \(D\): This stands for the "first derivative" or "differentiation of".
• \(D^2\): This stands for the "second derivative" or "differentiation of the differentiation".

In the equation \((D^2 - 1) y = \cos 2x\), applying the operator \(D^2 - 1\) to \(y\) means finding \(y'' - y\). Understanding this operation is crucial for simplifying and manipulating the equation while finding solutions. The differential operator helps us express complex differentiation operations more compactly and manageably.

A non-homogeneous linear differential equation includes an additional term or function that is not a derivative of the dependent variable. It can generally be represented in the form:

• Homogeneous Part: Terms involving derivatives of \(y\).
• Non-Homogeneous Part: An additional function that does not depend on \(y\).

In the equation \((D^2 - 1) y = \cos 2x\), the right-hand side \(\cos 2x\) makes it non-homogeneous. This necessitates finding a particular solution, often involving assuming a form similar to the non-homogeneous term. In our case, since \(\cos 2x\) is a trigonometric function, it signals that our assumed solution should also include trigonometric terms like \(A\cos 2x + B\sin 2x\). Once this particular solution is found, verifying it ensures the equation balances out.

Trigonometric functions like \(\cos(x)\) and \(\sin(x)\) frequently appear in differential equations, especially when the solution involves oscillatory or wave-like behaviors. In our exercise, the presence of \(\cos 2x\) as part of the equation suggests the assumed form of the particular solution must include similar functions. Here's why:

• Cosine Function: \(\cos(2x)\) indicates repeated cycles every \(\pi\) radians due to its periodic nature.
• Sine Function: \(\sin(2x)\) complements cosine, forming a complete basis for oscillatory functions.

By assuming a solution of the form \(y_p = A\cos 2x + B\sin 2x\), we are leveraging these functions' properties to easily handle differential and integral calculations, especially since they convert easily under differentiation. This makes them highly useful when forming solutions that need to match or cancel specific trigonometric terms in the differential equation.

The process of computing derivatives is central when working with differential equations. Here, we specifically deal with finding derivatives of trigonometric functions.

• First Derivative: For \(y_p = A\cos 2x + B\sin 2x\), the first derivative is obtained as \(y_p' = -2A\sin 2x + 2B\cos 2x\). This follows from the standard differentiation rules for sine and cosine.
• Second Derivative: The second derivative is \(y_p'' = -4A\cos 2x -4B\sin 2x\). Note how trigonometric functions easily swap under differentiation, making it straightforward to find these terms.

Once these derivatives are calculated, they are substituted back into the differential operator part
\((D^2 - 1)\) to check if the solution satisfies the original equation. Understanding derivative computation enables us to accurately match the form and solve for specific constants that make our equation true.