Problem 70 - Calculus Volume 3

Question

a. Write the form for the particular solution \(y_{p}(x)\) for the method of undetermined coefficients. b. Use a computer algebra system to find a particular solution to the given equation. $$ 4 y^{\prime \prime}+5 y^{\prime}-2 y=e^{2 x}+x \sin x $$

Step-by-Step Solution

Answer

Use the form \( y_p(x) = Ae^{2x} + (Bx+C)\cos x + (Dx+E)\sin x \) and solve using a CAS.

Step 1: Identify the Non-Homogeneous Terms

The given differential equation is \(4y'' + 5y' - 2y = e^{2x} + x \sin x \). The non-homogeneous terms on the right side are \(e^{2x}\) and \(x \sin x\). Each term will contribute to the form of the particular solution.

Step 2: Write the Form for \(y_{p}(x)\) using Undetermined Coefficients

For \(e^{2x}\), the chosen form is \(Ae^{2x}\). For \(x \sin x\), the given expression is the product of a polynomial and a sine function, so the chosen form is \((Bx + C)\cos x + (Dx + E)\sin x\). Thus, the particular solution form is:\[y_{p}(x) = Ae^{2x} + (Bx + C)\cos x + (Dx + E)\sin x\]

Step 3: Use a Computer Algebra System (CAS)

Use a Computer Algebra System (CAS) like Mathematica, MATLAB, or Maple to substitute \(y_p(x)\) and its derivatives back into the differential equation and solve for the coefficients \(A, B, C, D,\) and \(E\).

Step 4: Evaluate and Confirm the Particular Solution

By solving the system of equations generated by equating coefficients, the CAS will determine the values of parameters \(A, B, C, D,\) and \(E\). The specific values will give you the particular solution.

Key Concepts

Non-Homogeneous Differential Equation

A non-homogeneous differential equation is an equation that has terms without the dependent variable or its derivatives. These terms on the right-hand side make the differential equation non-simple, meaning they add an extra twist. The key feature of a non-homogeneous equation is this additional term, which distinguishes it from a homogeneous equation where the right side is zero.

When dealing with non-homogeneous differential equations, our goal is to find a solution that includes both a complementary solution (related to the homogeneous part) and a particular solution (related to the non-homogeneous part). In our given exercise, the non-homogeneous terms are the expressions such as \( e^{2x} \) and \( x \sin x \) on the right side of the equation. These terms are not derived from the differential equation itself, thus giving it the 'non-homogeneous' designation. Understanding how these terms affect the solution is crucial for applying methods like undetermined coefficients.

Particular Solution

The particular solution is a specific part of the overall solution to a non-homogeneous differential equation. It caters specifically to the non-homogeneous terms. Unlike the complementary solution, which solves the associated homogeneous equation, the particular solution addresses those pesky terms that differ from zero on the equation's right-hand side.

To find the particular solution, we often use a method called "undetermined coefficients." This method requires you to guess the form of the solution based on the non-homogeneous terms and then determine unknown coefficients so that the solution fits the equation. In our example, we guessed a form \( y_{p}(x) = Ae^{2x} + (Bx + C)\cos x + (Dx + E)\sin x \). This process involves guessing the terms as functions, such as exponential or trigonometric, matching those present in the equation. It's a bit like solving a puzzle where the shape hints at which pieces to try.

Computer Algebra System

A Computer Algebra System (CAS) is a software program that automates calculations, symbolic representations, and algebraic manipulations. These handy tools, like Mathematica, MATLAB, or Maple, are invaluable for solving complex mathematical problems, including differential equations.

In our exercise, the CAS simplifies the tedious work of finding the particular solution by substituting \( y_{p}(x) \) and its derivatives back into the differential equation. It computes the coefficients \( A, B, C, D, \) and \( E \) quickly and accurately, which might be cumbersome to do by hand. This efficiency allows students and mathematicians to focus on analyzing and understanding problems rather than being bogged down by arithmetic. Using a CAS is like having a superpower that takes care of the heavy lifting, so you can focus on strategy and insights.