6.3. Finding the Particular Integral

There are multiple methods for finding y (x). The most common ones used in engineering applications are:

In engineering, when working with non-homogeneous differential equations, there are key methods to find the particular integral (PI). These methods allow for the solution of equations when external forces are applied to a system.

Think of it like trying to heat a pot of water on a stove. The heat from the stove can be compared to the external forces acting on a system, and the way you monitor the water's temperature is similar to how engineers find the responses in differential equations.

6.3.1 Method of Undetermined Coefficients

When to Use

• Only when f(x) is a linear combination of functions like:
  • Polynomials (e.g. x, x²)
  • Exponentials (e.g. e^ax)
  • Trigonometric functions (e.g. sinx, cosx)
  • Their products (e.g. x e^x, e^x cosx, etc.)

Procedure

1. Guess a form for y (x), based on the form of f(x), with unknown coefficients.
2. Substitute this guess into the differential equation.
3. Determine the coefficients by matching both sides.

Important Note – Modification Rule If the guessed form for y is a solution of the homogeneous part (i.e., appears in y_h), multiply by x or a higher power of x until linear independence is achieved.

The Method of Undetermined Coefficients helps solve differential equations when the non-homogeneous function f(x) is in a specific format. First, you make an educated guess about the form of y_p that mirrors the structure of f(x). After guessing, you substitute this guess into the differential equation. Then, by matching coefficients on both sides of the equation, you can find the unknowns in your guess. If your guess is just like the solution from the homogeneous part, you modify it by multiplying it by x to ensure it’s unique.

Imagine a student trying to solve a math problem by identifying the pattern in similar previous problems they solved. If they realize that part of their solution looks too familiar because it’s the same as a previous problem, they tweak their answer by adding an extra variable (like multiplying by x) to ensure it’s unique.

Example 1: Solve:

d²y/dx² - 3dy/dx + 2y = e^x Solution:

• Auxiliary Equation: r² - 3r + 2 = 0 ⇒ r = 1, 2
• y_h = C₁e^x + C₂e²x
• Since e^x already appears in y_h, guess y_p = Ax e^x.
• Differentiate and substitute into the original equation.
• Find A, and then write the complete solution: y = C₁e^x + C₂e²x + Ax e^x

In this example, we start with a non-homogeneous equation where the right side is e^x. The first step is finding the homogeneous solution (y_h) using its auxiliary equation. Since e^x is already in the y_h, we use a new guess for the particular integral y_p, which is Ax e^x. By substituting this guess back into the original equation, we can solve for A and obtain the complete solution by adding y_h and y_p together.

Consider tuning a musical instrument. You start with a basic sound but realize your original string is already in tune (the homogeneous solution), so you add another layer of sound (the Ax e^x) to enrich it. Once you match the new sound’s pitch with the existing notes, you have a harmonious tune (the complete solution).

6.3.2 Method of Variation of Parameters

When to Use

• When f(x) is not of the standard type or not suitable for undetermined coefficients.
• Works for any form of f(x), but involves integration.

Procedure Given:

• ay′′ + by′ + cy = f(x)

1. Solve the homogeneous equation to find y₁(x), y₂(x), the two linearly independent solutions.
2. Assume: yₚ(x) = u₁(x)y₁(x) + u₂(x)y₂(x)
3. Find u₁(x), u₂(x) by solving the system: u₁y₁ + u₂y₂ = 0 u₁′y₁ + u₂′y₂ = f(x)/(a)
4. Integrate u₁(x), u₂(x) to get u₁', u₂', then substitute into yₚ(x).

The Method of Variation of Parameters is applicable when the function f(x) does not fit into simple forms appropriate for the previous method. Instead of guessing a specific function form, this method assumes that the particular integral is a combination of the independent solutions of the homogeneous equation, modified by functions u₁ and u₂. The next step is solving a system of equations to obtain these u functions, which are then integrated to finalize the particular integral.

This method is akin to customizing a recipe where the base recipe is known (the homogeneous solutions), and you want to add new ingredients (the particular integral) that suit your taste. Instead of just adding any ingredient, you need to ensure it complements the base recipe by solving for what works best while keeping the dish balanced and delicious.

Example

Solve: y′′ + y = tan(x) Solution:

• Homogeneous solution: y_h = Ccos(x) + Csin(x)
• Let y₁ = cos(x), y₂ = sin(x) Using variation of parameters:
• u₁(x)cos(x) + u₂(x)sin(x) = 0
• -u₁'(x)sin(x) + u₂'(x)cos(x) = tan(x)
• Solve these equations to find u₁, u₂, integrate, and get the particular integral yₚ.

In this example, we are asked to solve a differential equation where the right side is tan(x). We begin by finding the homogeneous solution. Then using the Method of Variation of Parameters, we set up equations with u₁ and u₂. By solving these equations, we find u functions, which we integrate to obtain the specific solution needed to solve the full equation.

Picture a tailor sewing a suit. They use a predefined pattern (the homogeneous solution) but need to adjust the fit with custom measurements (u₁ and u₂). Instead of using one size fits all, they ensure the suit fits perfectly based on the unique characteristics of the person wearing it. Once adjusted, the suit is ready for wear (the solution to the equation).

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