8.3. Method of Solving: Auxiliary Equation Method

To solve such equations, we use the operator method (also known as the symbolic method) where differential operators 𝐷 = ∂ and 𝐷′ = ∂ are used. ∂𝑥 ∂𝑦

The Auxiliary Equation Method is a systematic technique used to solve homogeneous linear partial differential equations (PDEs) with constant coefficients. We start by defining differential operators, where 𝐷 represents differentiation with respect to x (∂/∂x) and 𝐷′ represents differentiation with respect to y (∂/∂y). This helps to simplify the equations into a more manageable form that can be tackled using algebraic methods.

Think of the operator method as a set of tools or instruments that simplify the process of assembling a piece of furniture. Just as a screwdriver and a wrench help you manipulate screws and bolts, the differential operators allow us to manipulate the derivatives in the equations lightly.

Example: ∂2𝑧 ∂2𝑧 ∂2𝑧 +2 + = 0 ∂𝑥2 ∂𝑥∂𝑦 ∂𝑦2 can be written as: (𝐷2+2𝐷𝐷′ +𝐷′2 )𝑧 = 0

In this step, we take the given PDE and rewrite it in operator form using the defined differential operators. For example, the PDE that involves second-order derivatives is transformed using 𝐷 and 𝐷′. This conversion simplifies the equation, allowing us to handle it more easily in the next steps.

Imagine converting a recipe that uses various cooking terms into a simpler list of instructions. By eliminating jargon (like converting 'simmer' to 'keep warm'), you can follow the steps more straightforwardly. Similarly, converting the PDE to operator form helps clarify the next steps involved.

Replace 𝐷 →𝑚 and 𝐷′ → 1 to form an algebraic equation: 𝑚2 + 2𝑚+1 = 0 ⇒ (𝑚+ 1)2 = 0 This is called the Auxiliary Equation.

In this step, the differential operators in the operator form are substituted with algebraic variables. We replace 𝐷 with 'm' and 𝐷′ with '1', resulting in an algebraic equation known as the Auxiliary Equation (AE). This equation plays a crucial role in determining the solutions of the original PDE by revealing the roots, which guide the form of the general solution.

Consider a detective trying to solve a mystery by turning complex clues into simple leads. By converting clues into manageable pieces of information (like formulas in math), the detective finds a clear path to the solution. Similarly, forming the Auxiliary Equation provides clarity to solve the PDE.

Solve the AE for 𝑚. The nature of roots (distinct, repeated, complex) will guide the form of the solution.

Once we have the Auxiliary Equation, we solve it for the variable 'm' to find its roots. These roots can be distinct, repeated, or complex, and each type influences how we will write the solution to the original PDE. This step is essential because the characteristics of the roots will indicate the structure of the complementary function in the general solution.

Think of the roots of the AE like the keys to a locked door. Each type of key (distinct, repeated, complex) has a different impact on how you open the door (finding the solution to the PDE). Depending on the key you have, you might open it in different ways.

Based on the roots: • Distinct real roots 𝑚1,𝑚2: 𝑧 = 𝑓1(𝑦−𝑚1𝑥)+𝑓2(𝑦 −𝑚2𝑥) • Repeated root 𝑚: 𝑧 = 𝑓1(𝑦−𝑚𝑥)+𝑥𝑓2(𝑦−𝑚𝑥) • Complex roots 𝛼 ±𝑖𝛽: 𝑧 = 𝑓1(𝑦 −𝛼𝑥)cos(𝛽𝑥)+ 𝑓2(𝑦 −𝛼𝑥)sin(𝛽𝑥)

This step involves constructing the complementary function (CF) based on the types of roots obtained from the Auxiliary Equation. If the roots are distinct real numbers, we combine functions of these roots. For repeated roots, we introduce an additional linear term, and for complex roots, we incorporate sine and cosine functions. Each case reflects how the solution space of the PDE is structured.

Imagine building a house with different types of materials (wood for distinct roots, metal for repeated roots, and bricks for complex roots). Depending on the materials you have, your house will look and function differently. In the same way, the solution structure of the PDE changes based on the roots of the AE.

In this topic, we focus only on homogeneous equations, so Particular Integral (PI) is not required.

In this particular method, we only address homogeneous equations, meaning all terms are linked through the dependent variable and its derivatives without any additional external forces or 'free' terms. Thus, we do not need to compute a Particular Integral, which would be necessary if we were dealing with non-homogeneous equations.

Think of a cake recipe that strictly calls for certain ingredients without any extra toppings; those toppings are like non-homogeneous terms. By sticking to the basic ingredients (the homogeneous part), we simplify the recipe, ensuring our focus remains on the core cake requirements.

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