91Ó°ÊÓ

Second-order linear differential equations are mathematical expressions that involve a function and its derivatives up to the second derivative. These equations appear in the form \( a y'' + b y' + c y = f(x) \), where \( a \), \( b \), and \( c \) are constants, and \( f(x) \) is a function of \( x \). In our case, the equation is \( y'' + 25y = 20 \sin 5x \). Here, the second-order derivatives play a crucial role in modeling dynamic systems such as mechanical vibrations, electrical circuits, and many other physical phenomena.

These equations can be categorized into two main types:

• Homogeneous equations: When the right-hand side \( f(x) \) is zero, leading to a general form like \( a y'' + b y' + c y = 0 \).
• Non-homogeneous equations: When \( f(x) eq 0 \), contributing additional complexity to the solutions.

The focus in solving these equations is typically to find the particular and homogeneous solutions and combine them into a general solution.

The characteristic equation is a tool used to solve homogeneous linear differential equations. It converts the differential equation into an algebraic equation that is easier to handle. For the homogeneous equation \( y'' + 25y = 0 \) in our problem, the characteristic equation is found by assuming the solution has the form \( y = e^{rt} \).

Substitute \( y = e^{rt} \) into the homogeneous equation, yielding:

• \( r^2 + 25 = 0 \)

The roots of this algebraic equation, \( r = \pm 5i \), indicate the nature of the solutions of the differential equation. Here, the purely imaginary roots suggest oscillatory solutions comprised of sine and cosine functions, leading to the homogeneous solution: \( y_h = C_1 \cos 5x + C_2 \sin 5x \). These forms are typical when dealing with complex roots in second-order differential equations.

A non-homogeneous differential equation includes a term that is not dependent on the function or its derivatives, for example, \( 20 \sin 5x \) in our case. Solving such equations involves finding a particular solution, which accounts for this extra term.

To solve:

• Identify the non-homogeneous part: Here, it's \( 20 \sin 5x \).
• Propose a particular solution: We propose \( y_p = Ax \cos 5x + B \sin 5x \), considering the similarities with the homogeneous solution.

The particular solution must then satisfy the original non-homogeneous equation. The method of undetermined coefficients helps in determining values for \( A \) and \( B \) by substituting into the differential equation and matching coefficients. After finding the suitable values, these contribute to the construction of this particular solution, which is essential to form the complete general solution when combined with the homogeneous solution.

A homogeneous solution is a part of the complete solution for differential equations. It is derived from the homogeneous form of the given differential equation where the right-hand side is set to zero. The homogeneous equation in our problem is \( y'' + 25y = 0 \).

To find the homogeneous solution:

• Solve the characteristic equation, \( r^2 + 25 = 0 \), giving roots \( r = \pm 5i \).
• These roots signify oscillations, translating to the trigonometric functions in the solution.
• The homogeneous solution thus takes the form \( y_h = C_1 \cos 5x + C_2 \sin 5x \).

This solution represents the natural response of the differential equation system without the external force or non-homogeneous term. It forms the base upon which the particular solution is added to capture the full behavior of the solution, acknowledging both natural dynamics and the enforced alterations due to the non-homogeneous part.