91Ó°ÊÓ

A second-order linear differential equation is an equation that involves the second derivative of a function, typically denoted as \( y'' \), and it is of the form:
\[ a y'' + b y' + c y = g(t) \]This means the highest derivative is the second derivative (hence second-order), and linear because the function and its derivatives appear to the first power.
For example, the given equation \( y'' + 9y = 3\sin(3t) \) is second-order because it involves \( y'' \), and linear because both \( y'' \) and \( y \) are linear terms. The non-homogeneous part here is represented by \( 3\sin(3t) \). Understanding the order of a differential equation helps us determine the complexity of the solution and the methods required.
To solve it, you always need to find both the homogeneous solution and a particular solution, adding them together to get the general solution.

The homogeneous solution deals with solving the equation when the non-homogeneous part or function \( g(t) \) is zero. For the given differential equation, it involves solving:
\[ y'' + 9y = 0 \]
To find the homogeneous solution, we use an approach that involves assuming a solution of the form \( y = e^{rt} \), leading to the characteristic equation:

• \( r^2 + 9 = 0 \)

After solving, we find the roots as \( r = \pm 3i \), indicating complex roots. Therefore, the general solution to this homogeneous equation is expressed using trigonometric functions:

• \( y_h = c_1 \cos(3t) + c_2 \sin(3t) \)

The \( c_1 \) and \( c_2 \) are constants determined by initial conditions, if given. This solution describes a system that only depends on the characteristic equation, without any influence from external forces.

A non-homogeneous differential equation includes a non-zero function on the right side, such as:
\[ y'' + 9y = 3 \sin(3t) \]
This equation has a part \( g(t) = 3\sin(3t) \), making it non-homogeneous. To solve it, you first find the homogeneous solution, solving the equation as if \( g(t) \) was zero.
Then, locate a particular solution \( y_p \) that fits the form of \( g(t) \). A common technique is the method of undetermined coefficients, where you assume a form for \( y_p \):

• Example for \( 3\sin(3t) \): \( y_p = A t \cos(3t) + B t \sin(3t) \)

Substitute into the original equation and solve for the coefficients \( A \) and \( B \). In this case:

• \( A = 0 \), \( B = -\frac{1}{6} \)

This yields a particular solution of \( y_p = -\frac{1}{6}t \sin(3t) \), which addresses the influence of \( g(t) \).

The general solution of a second-order linear differential equation combines both the homogeneous and particular solutions. It provides a complete description of all potential solutions
For such problems, the general solution is formulated as:
\[ y = y_h + y_p \]This means you add the homogeneous solution \( y_h \) and the particular solution \( y_p \).

• Homogeneous solution: \( c_1 \cos(3t) + c_2 \sin(3t) \)
• Particular solution: \( -\frac{1}{6}t \sin(3t) \)

Combining these, the comprehensive solution becomes \( y(t) = c_1 \cos(3t) + (c_2 - \frac{1}{6}t) \sin(3t) \).
The constants \( c_1 \) and \( c_2 \) are determined by specific initial values or boundary conditions, providing solutions adapted to particular scenarios or physical problems.