91Ó°ÊÓ

Finding a particular solution to a differential equation involves identifying a specific solution that fits the non-homogeneous equation. In this context, the differential equation is given by \( y'' + y = b(x) \). The particular solution \( \psi_{p}(x) \) is proposed as \( \int_{1}^{x} \sin(x-t) b(t) \, dt \). To verify this, one must substitute \( \psi_{p}(x) \) back into the equation. This involves computing the necessary derivatives and ensuring that the left-hand side becomes \( b(x) \) when the particular solution and its derivatives are substituted. This specific function \( \psi_{p} \) effectively "neutralizes" the non-homogeneous part \( b(x) \), showing it to be a valid particular solution.

In the context of differential equations, a solution is said to be bounded if it does not become infinite as \( x \) approaches infinity. For the equation \( y'' + y = b(x) \), the task is to show that all solutions are bounded on the interval \( 1 \leq x < \infty \). In solving a differential equation, the general solution typically includes a homogeneous solution part plus any particular solution. The homogeneous solution is of the form \( C_1 e^x + C_2 e^{-x} \). For boundedness, any terms that could lead to infinity, such as \( e^x \), must be ruled out by setting their coefficients to zero. The solution thus simplifies, and focus shifts to proving that the particular solution \( \psi_{p}(x) \) remains bounded. Given the condition that \( \int_{1}^{\infty} |b(t)| \, dt < \infty \), it implies that the area under the curve of \( b(t) \) is finite, supporting the conclusion that \( \psi_{p}(x) \) remains finite as \( x \to \infty \).

Verification involves checking that a proposed solution satisfies the given differential equation. This is done by substituting the solution and its derivatives into the equation to see if they simplify correctly back to the function on the right-hand side, \( b(x) \) in this case. For \( \psi_{p}(x) = \int_{1}^{x} \sin(x-t) b(t) \, dt \), the verification process involves a few key steps:

• Calculate the first and second derivatives of \( \psi_{p}(x) \).
• Substitute these derivatives into the left-hand side of the differential equation \( y'' + y \).
• Confirm that the left-hand side equals \( b(x) \).

Once this equality is confirmed, it verifies that \( \psi_{p}(x) \) is a legitimate particular solution of the equation.

Continuous functions play a fundamental role in differential equations, especially when dealing with integrals and the behavior of solutions over specific intervals. In this exercise, the function \( b(x) \) must be continuous on the interval \( 1 \leqq x < \infty \). Continuity ensures that the behavior of \( b(x) \) is smooth, with no jumps or discontinuities causing issues in integration. Another important aspect of continuity here is that:

• The given condition \( \int_{1}^{\infty} |b(t)| \, dt < \infty \) relies on \( b(x) \) being continuous, guaranteeing a well-defined integral.
• This condition also implies the function is absolutely integrable over the given interval.

These properties are crucial in showing that a solution to the differential equation remains bounded, as it allows the particular solution based on \( b(x) \) to be finite for all \( x \) in the specified domain.