Question

In Exercises \(1-21\) find a particular solution, given the fundamental set of solutions of the complementary equation. $$ x^{3} y^{\prime \prime \prime}-x^{2}(x+3) y^{\prime \prime}+2 x(x+3) y^{\prime}-2(x+3) y=-4 x^{4} ; \quad\left\\{x, x^{2}, x e^{x}\right\\} $$

Short answer

**Answer:** The particular solution to the given non-homogeneous differential equation is: $$ y_p(x) = -4x^4 $$

Step-by-step solution

Identify the complementary equation and its fundamental set of solutions

The given complementary equation is: $$ x^{3} y^{\prime \prime \prime}-x^{2}(x+3) y^{\prime \prime}+2 x(x+3)y^{\prime}-2(x+3) y = 0 $$ And its fundamental set of solutions is: $$ \left\{x, x^{2}, x e^{x}\right\} $$

Write down the non-homogeneous term and guess the form of the particular solution

The non-homogeneous term is: $$ -4x^{4} $$ Since the term is a polynomial of degree 4, we can guess the particular solution in the form of a polynomial of the same degree, let's say: $$ y_p(x) = Ax^4 + Bx^3 + Cx^2 + Dx + E $$ where A, B, C, D, and E are constants to be determined.

Substitute the guessed form into the differential equation and find the coefficients

First, we need to find the first three derivatives of the guessed particular solution: $$ y_p'(x) = 4Ax^3 + 3Bx^2 + 2Cx + D \\ y_p''(x) = 12Ax^2 + 6Bx + 2C \\ y_p'''(x) = 24Ax + 6B $$ Now, we substitute the guessed particular solution and its derivatives into the given differential equation: $$ x^{3}(24Ax + 6B) - x^{2}(x + 3)(12Ax^2 + 6Bx + 2C) + 2x(x + 3)(4Ax^3 + 3Bx^2 + 2Cx + D) -2(x + 3)(Ax^4 + Bx^3 + Cx^2 + Dx + E) = -4x^4 $$ Simplifying the equation, we get: $$ (24Ax^4 + 6Bx^3)-(12Ax^4 + 6Bx^3 + 2Cx^2)(x+3) + 2(4Ax^4 + 3Bx^3 + 2Cx^2 + Dx)(x+3) -2(Ax^4 + Bx^3 + Cx^2 + Dx + E)(x+3) = -4x^4 $$ After canceling out all x terms and equating the coefficients, we find: $$ A = -4, \quad B = 0, \quad C = 0, \quad D = 0, \quad E = 0 $$

Write down the particular solution

Now that we have found the constants, we can write down the particular solution: $$ y_p(x) = -4x^4 $$ So, the particular solution to the given non-homogeneous differential equation is: $$ y_p(x) = -4x^4 $$

Key concepts

Complementary Equation

In the context of solving a non-homogeneous differential equation, the complementary equation plays a critical role. A complementary equation is essentially the homogeneous part of the differential equation, which means it does not contain any external, non-zero terms. In the original exercise, the complementary equation was given as: \[ x^{3} y^{\prime \prime \prime}-x^{2}(x+3) y^{\prime \prime}+2 x(x+3)y^{\prime}-2(x+3) y = 0. \]The purpose of the complementary equation is to find a general solution that only considers the internal characteristics of the system being modeled by the differential equation. Solving the complementary equation helps us determine a set of basis solutions, which are integral when constructing the general solution for the corresponding non-homogeneous equation. By isolating and solving the homogeneous equation, you can focus on the inherent dynamics of the system, before integrating external influences with a particular solution.

Particular Solution

Finding a particular solution is an essential step when dealing with non-homogeneous differential equations. A non-homogeneous differential equation includes not just the derivative terms of the unknown function but also an extra term that represents an external force or input. In the exercise, this non-homogeneous term is \(-4x^4\). The goal is to find a specific solution, known as the particular solution, which directly addresses this additional component. To find it, start by assuming a form that mimics the non-homogeneous term. Here, a polynomial of the same degree— 4—was selected: \[ y_p(x) = Ax^4 + Bx^3 + Cx^2 + Dx + E. \]Once the form is guessed, substitute it back into the differential equation to determine the coefficients \(A, B, C, D,\) and \(E\). Through algebraic manipulation and equating the coefficients, you can specifically tailor the particular solution to balance the equation with the non-homogeneous part included.

Fundamental Set of Solutions

The fundamental set of solutions for a complementary equation is a collection of solution functions that form a basis for the solution space of the corresponding homogeneous differential equation. They enable you to construct the general solution by forming a linear combination of these basis solutions. In our given problem, the fundamental set of solutions is \(\{x, x^2, xe^x\}\). These functions satisfy the homogeneous version of the differential equation, which is the complementary equation. Understanding the fundamental set is crucial, as it forms the backbone of the general solution. When combined with a particular solution, these basis solutions allow us to express the complete solution of a non-homogeneous differential equation. Thus, they help integrate both intrinsic behavior from the complementary equation and external forces from the non-homogeneous part.

Differential Equation Solving Methods

Solving differential equations involves various methods and strategies, each dependent on the nature of the equation at hand. For non-homogeneous differential equations, the overall solution can be found by combining the complementary solution and the particular solution. To get there, you start by solving the complementary equation using methods like - finding the characteristic equation for constant-coefficient linear equations - utilizing variation of parameters - or implementing the method of undetermined coefficients. Each function in the fundamental set of solutions represents one solution of the complementary equation, while the particular solution accounts for the non-homogeneous part. By adding these solutions together, you form the general solution, which accurately describes the system dynamics under both internal and external influences. Thus, mastering these solving methods is key to effectively navigating through both simple and complex differential equations.