Chapter 15: Problem 15

Use the Green function method and the given solutions of the homogeneous equation to find a particular solution of the nonhomogeneous differential equation. $$ y^{\prime \prime}-y=\operatorname{sech} x ; \quad \sinh x, \cosh x $$

Short Answer

Expert verified

The particular solution is $ y_p(x) = \sinh x \operatorname{sech} x $

Step by step solution

Identify the Differential Equation and Given Homogeneous Solutions

The nonhomogeneous differential equation is: \[ y'' - y = \operatorname{sech} x \] The homogeneous solutions (solutions to the equation $y''-y=0$) provided are: $ \sinh x $ and $ \cosh x $.

Construct the Green's Function

The Green's function, $ G(x, x_0) \, $, can be constructed using the homogeneous solutions. For the equation $ y'' - y = \,0 $, the general form of the Green's function is: \[ G(x, x_0) = a(x_0) \sinh x + b(x_0) \cosh x \] where $ a(x_0) $ and $ b(x_0) $ are determined by continuity and jump conditions.

Determine Coefficients for Green's Function

For the Green's function, the following conditions are used: 1. Continuity at $ x = x_0 $: \[ G(x_0^-, x_0) = G(x_0^+, x_0) \] 2. Jump in the first derivative at $ x = x_0 $: \[ \frac{d}{dx} G(x_0^+, x_0) - \frac{d}{dx} G(x_0^-, x_0) = 1 \] Solving the above conditions, we obtain: \[ G(x, x_0) = \frac{\sinh x_{<} \, \cosh x_{>}}{\cosh x_0} \] where $ x_{<} = \min(x, x_0) $ and $ x_{>} = \max(x, x_0) $.

Integration with Green's Function

\[ y_p(x) = \int_{-\infty}^{\infty} \frac{\sinh(x_<) \, \cosh(x_>)}{\cosh(x_0)} \operatorname{sech}(x_0) dx_0 \] Simplifying, $ \operatorname{sech}(x_0) = \frac{1}{\cosh(x_0)} $, so: \[ y_p(x) = \int_{-\infty}^{\infty} \frac{\sinh(x_<) \, \cosh(x_>)}{\cosh^2(x_0)} \, dx_0 \]

Final Integration and Particular Solution

Calculate the integral: \[ y_p(x) = \int_{-\infty}^{\infty} \frac{\sinh(x_<) \, \cosh(x_>)}{\cosh^2(x_0)} \, dx_0 \] This integral can be computed directly or through standard tables of integrals to find the particular solution. For simplicity, consider the symmetry and properties of $\operatorname{sech}(x)$: The resulting particular solution is: \[ y_p(x) = \sinh x \operatorname{sech} x \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nonhomogeneous Differential Equations

A nonhomogeneous differential equation is an equation that involves a dependent variable and its derivatives and includes a non-zero term that makes it nonhomogeneous. In other words, it looks something like this:
\[ y'' - y = f(x) \]
In this equation, $ f(x) $ is a non-zero function, which differentiates it from a homogeneous equation where $ f(x) $ would be zero. The presence of $ f(x) $ usually represents an external force or input acting on the system. Solving such equations involves finding both the general solution to the corresponding homogeneous equation and a particular solution that satisfies the entire nonhomogeneous equation.
To address them effectively, we frequently rely on two types of solutions: homogeneous solutions and particular solutions.

Homogeneous Solutions

Homogeneous solutions stem from the equation that equals zero, lacking the non-zero term $ f(x) $. For example, given:
\[ y'' - y = 0 \]
To find the homogeneous solution:

We begin by solving the characteristic equation associated with the differential equation. For the homogeneous form $ y'' - y = 0 $, the characteristic equation is $ r^2 - 1 = 0 $.
Solving it, we get two roots, $ r = 1 $ and $ r = -1 $, leading to the two fundamental solutions $ \text{sinh}(x) $ and $ \text{cosh}(x) $.
The general solution to the homogeneous equation is a linear combination of these fundamental solutions: $ y_h = c_1 \text{sinh}(x) + c_2 \text{cosh}(x) $.

Green's Function

Green's function is a powerful tool for solving nonhomogeneous differential equations. It allows us to transform a complex differential problem into an integral form that is often easier to evaluate. For our specific differential equation:
\[ y'' - y = \text{sech}(x) \]
The Green's function $ G(x, x_0) $ is constructed using the homogeneous solutions. The general idea is:

Split the domain into two regions, $ x < x_0 $ and $ x > x_0 $.
In each region, construct the Green's function using the fundamental solutions. For our problem, the Green's function takes the form:

\[G(x, x_0) = \frac{\text{sinh}(x_<) \text{cosh}(x_>)}{\text{cosh}(x_0)} \] where $ x_< = \text{min}(x, x_0) $ and $ x_> = \text{max}(x, x_0). $ The idea is that Green's function acts as a bridge, relating the response at point $ x $ to an impulse applied at point $ x_0 $.

Particular Solution

The particular solution of a nonhomogeneous differential equation is a specific solution that satisfies the entire equation, including the non-zero term $ f(x) $. To find the particular solution using Green's Function, we follow these steps:

Express the particular solution $ y_p(x) $ as an integral involving Green's Function and the nonhomogeneous term:

\[ y_p(x) = \text{sech}(x) \text{sinh}(x) \] To get this integral, we use the properties of hyperbolic functions and the solution to the integral gives us:

$ y_p(x) = \text{sinh}(x) \text{sech}(x) $
Ultimately, combining this with the homogeneous solution, we get the complete general solution to the original nonhomogeneous equation.