91Ó°ÊÓ

When dealing with differential equations, one crucial step is finding a particular solution. This is separate from the complementary solution, which solves the homogeneous version of an equation. The particular solution specifically addresses the non-homogeneous part. To create this solution, we conjecture a form that resembles the non-homogeneous term on the right side of the equation.

In this instance, our differential equation is given by:

• The non-homogeneous term: \( \sin x + x \cos x \)

Using our previous knowledge of related functions, we use a guessed form like:

• \( y_p = A x \cos x + B x \sin x + C \sin x \)

Here, \(A\), \(B\), and \(C\) are constants we need to determine. By substituting this guess into the differential equation and equating coefficients, we can find these constants. This gives us the particular solution that complements the homogeneous solution to form the general solution of the differential equation.

A homogeneous equation is a fundamental concept in solving differential equations. It is obtained by eliminating the non-homogeneous part of the given differential equation. Doing so leaves us with an equation that only involves the derivatives of the function.

• A homogeneous equation helps in finding the complementary solution.

In our exercise, the homogeneous equation derived was:

• \( y^{\prime\prime} + y = 0 \)

To solve this, we identified the characteristic equation:

• \( r^2 + 1 = 0 \)

Solving the characteristic equation gave us roots: \( r = \pm i \), leading to our complementary solution:

• \( y_c = C_1 \cos x + C_2 \sin x \)

This means any solution to the homogeneous equation will be a combination of sine and cosine functions. By combining these results with the particular solution, we can find the full solution to the original non-homogeneous equation.

The Method of Undetermined Coefficients is a valuable strategy in differential equations for finding particular solutions to non-homogeneous equations. This method works well when the non-homogeneous part is a function where its derivatives can be expressed similarly, like polynomials, exponentials, sines, and cosines.

• Non-homogeneous term: \( \sin x + x \cos x \)
• Assumed form: \( y_p = A x \cos x + B x \sin x + C \sin x \)

After substituting this form and its derivatives into the differential equation, we match coefficients of similar terms to solve for \(A\), \(B\), and \(C\). Systems of equations are formed, and through quite straightforward algebraic manipulation, the constants can be determined.

• Given conditions: \( A = -1 \), \( B = 0 \), \( C = -1 \)

Using this approach, the guessed form turns into our particular solution. This method is a trial-and-error type approach where the right form needs to be initially guessed, but once the form is accurate, the solution process becomes quite efficient.