Problem 75 - 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. $$ y^{\prime \prime}+2 y^{\prime}-8 y=6 e^{2 x} $$

Step-by-Step Solution

Answer

The particular solution is \( y_p(x) = e^{2x} \).

Step 1: Identify the form of the non-homogeneous term

The non-homogeneous term in the differential equation is \( 6e^{2x} \). This suggests that we should consider exponential functions in our particular solution form.

Step 2: Choose the appropriate form for particular solution

Since the non-homogeneous term involves an exponential function \( e^{2x} \), our assumed form for the particular solution \( y_p(x) \) will be \( y_p(x) = Ae^{2x} \), where \( A \) is a constant that needs to be determined.

Step 3: Differentiate the assumed particular solution

Calculate the first and second derivatives of \( y_p(x) = Ae^{2x} \):- \( y_p'(x) = 2Ae^{2x} \)- \( y_p''(x) = 4Ae^{2x} \)

Step 4: Substitute into the differential equation

Substitute \( y_p(x) \), \( y_p'(x) \), and \( y_p''(x) \) back into the original differential equation:\[y'' + 2y' - 8y = 6e^{2x}\]Substitute the derivatives:\[4Ae^{2x} + 2(2Ae^{2x}) - 8(Ae^{2x}) = 6e^{2x}\]

Step 5: Simplify and solve for A

Simplify the left side of the equation:\[4Ae^{2x} + 4Ae^{2x} - 8Ae^{2x} = 6e^{2x}\]This simplifies to:\[0 = 6e^{2x}\]Add like terms and simplify:\[ 6A e^{2x} = 6 e^{2x} \] Solve for \( A \):\[ A = 1 \]

Step 6: Write the particular solution

Now that we have found \( A = 1 \), the particular solution is:\[ y_p(x) = e^{2x} \]

Key Concepts

Non-Homogeneous Differential Equation

A non-homogeneous differential equation is a type of differential equation that includes a non-zero term on the right side of the equation. These equations generally take the form:\[ y'' + a y' + b y = g(x) \]where \( g(x) \) is the non-homogeneous part that makes the equation differ from homogeneous ones. These terms can be polynomials, exponentials, or trigonometric functions. Solving these equations involves finding both the complementary and particular solutions. The complementary solution is obtained by solving the related homogeneous equation where \( g(x) = 0 \). It reflects how the system behaves without external influences. Meanwhile, the particular solution accounts for the effects of the non-homogeneous term. Understanding this distinction is crucial for correctly solving such differential equations.

Particular Solution

The particular solution is a specific solution to a non-homogeneous differential equation that includes the non-homogeneous term's influence. For the method of undetermined coefficients, the aim is to guess the form of this particular solution based on the type of \( g(x) \).- **Exponential Form**: If \( g(x) \) is an exponential function like \( e^{2x} \), assume a solution of the form \( y_p(x) = A e^{2x} \).- **Polynomial Form**: If \( g(x) \) is polynomial, say \( ax^n \), you would choose a polynomial form like \( A_n x^n + A_{n-1} x^{n-1} + \ldots + A_0 \).- **Trigonometric Form**: If \( g(x) \) involves cosine or sine, your assumption might resemble \( A \cos(x) + B \sin(x) \).After assuming the particular solution, substitute it and its derivatives back into the original equation. By solving the resulting equation for unknown constants (like \( A \)), you find the particular solution. The key is in choosing the right form and adjusting it if overlaps with the complementary solution occur.

Computer Algebra System

A Computer Algebra System (CAS) is a software or tool that assists in mathematical calculations that involve symbolic manipulations. A CAS can solve equations, perform algebra, complete calculus operations, and more—all symbolically. This makes them incredibly powerful for solving complex differential equations. When dealing with a non-homogeneous differential equation like the one in our problem, a CAS can: - **Simplify Equations**: Automatically expand and simplify expressions. - **Solve for Constants**: Efficiently calculate unknown coefficients when using methods like undetermined coefficients. - **Verify Solutions**: Quickly check if your manually derived particular and complementary solutions satisfy the original equation. Using a CAS can be particularly beneficial in validating your manual work and ensuring accuracy in steps prone to arithmetic errors. They serve as an excellent learning tool because they allow you to see each intermediate step clearly, reinforcing learning through visualization.