Question

Find a particular solution, given the fundamental set of solutions of the complementary equation. $$ x y^{\prime \prime \prime}-(x-3) y^{\prime \prime}-(x+2) y^{\prime}+(x-1) y=-4 e^{-x} ; \quad\left\\{e^{x}, e^{x} / x, e^{-x} / x\right\\} $$

Short answer

Based on the given fundamental set of solutions and the method of undetermined coefficients, the particular solution of the differential equation \(x y^{\prime\prime\prime} - (x-3) y^{\prime\prime} - (x+2) y^{\prime} + (x-1) y = -4e^{-x}\) is \(y_p(x) = -\frac{4}{3}e^{-x}\).

Step-by-step solution

Identify the Form of the Particular Solution

Since the inhomogeneous term is \(-4e^{-x}\), we can guess the particular solution to be of the form \(Ae^{-x}\) where \(A\) is an undetermined coefficient.

Plug the Particular Solution into the Differential Equation

Now, we plug the particular solution, \(y_p = Ae^{-x}\), and its derivatives into the given differential equation. Firstly, we'll find the derivatives of the particular solution: $$ y_p' = -Ae^{-x}; \quad y_p'' = Ae^{-x}; \quad y_p''' = -Ae^{-x} $$ Now, we substitute these derivatives into the given differential equation: $$ x(-Ae^{-x}) - (x - 3)(Ae^{-x}) - (x + 2)(-Ae^{-x}) + (x - 1)(Ae^{-x}) = -4e^{-x} $$

Solve for the Undetermined Coefficient

Now, we simplify the left-hand side of the equation, factoring out the common term \(Ae^{-x}\): $$ Ae^{-x}(-x - x + 3 + x + 2 + x - 1) = -4e^{-x} $$ Simplify further: $$ Ae^{-x}(3) = -4e^{-x} $$ Now divide both sides by \(3e^{-x}\) to solve for \(A\): $$ A = -\frac{4}{3} $$

Write the Particular Solution

Now that we've found the value of the undetermined coefficient \(A\), we can write the particular solution as: $$ y_p(x) = -\frac{4}{3}e^{-x} $$

Key concepts

Complementary Equation Solutions

When solving differential equations, one of the techniques involved is finding the complementary solution, which is a solution to the associated homogeneous differential equation. The homogeneous equation is derived by setting the inhomogeneous term (the right side of the original equation that does not contain the function we are solving for or its derivatives) to zero. This equation is called 'complementary' because it complements the particular solution to form the general solution to the inhomogeneous differential equation.

In the given exercise, the fundamental set of solutions to the complementary equation is \( \{e^{x}, e^{x} / x, e^{-x} / x\} \). These solutions are specific to the homogeneous equation obtained by setting the right side of the original third-order differential equation to zero. It’s crucial to first solve the complementary equation because it ensures that the particular solution to the inhomogeneous equation doesn't duplicate the form of any part of the complementary solution. In this way, each piece of the general solution contributes something unique to satisfying the original differential equation.

Method of Undetermined Coefficients

The method of undetermined coefficients is an approach used to find a particular solution to a non-homogeneous, or inhomogeneous, differential equation when the inhomogeneous term is a simple function, like a polynomial, exponential, sine, or cosine. It works on the principle of making an educated guess about the form of the particular solution, then determining the unknown coefficients by plugging this guess into the differential equation.

In the problem at hand, the inhomogeneous term is \(-4e^{-x}\), suggesting a guess for the particular solution of the form \(Ae^{-x}\), where \(A\) is the coefficient we need to find. By substituting the assumed particular solution and its derivatives into the original equation and simplifying, we can solve for \(A\). This method is efficient for linear differential equations with constant coefficients, where the non-homogeneous part is of the form mentioned above.

Solving Inhomogeneous Differential Equations

Inhomogeneous differential equations contain a term that acts as an external force or input to the system described by the equation. Unlike homogeneous equations, which only consider the system's inherent properties, inhomogeneous equations account for external factors that can influence the system's behavior.

To solve such equations, we typically combine the complementary (or homogeneous) solution with the particular solution. The complementary solution is responsible for capturing the dynamic behavior due to the initial conditions and internal properties of the system, while the particular solution specifically addresses the inhomogeneous part. This is why, after finding the complementary solution set, we focused on seeking a particular solution that involves the term \(-4e^{-x}\). Combining these two parts gives us the general solution to the inhomogeneous differential equation.

Third-Order Differential Equation

Differential equations are categorized by the highest derivative present in the equation. A third-order differential equation involves the third derivative of the unknown function. They are more complex than first or second-order equations and can describe systems with more intricate dynamical behaviors.

In our exercise, the third-order differential equation is given by \(x y^{\prime \prime \prime}-(x-3) y^{\prime \prime}-(x+2) y^{\prime}+(x-1) y=-4 e^{-x}\). The presence of the variable \(x\) alongside the derivatives indicates that the equation is not only third-order but also non-linear, which increases the challenge in solving. Despite this complexity, we can still find a particular solution using the method of undetermined coefficients, as demonstrated in the exercise. When the fundamental set of solutions to the complementary equation is known, as in the problem statement, this provided starting point can guide us in determining the structure of the particular solution.