Question

Find the particular integral of each of the following: \((a) y^{(\prime \prime t)}(t)+2 y^{\prime \prime}(t)+y^{\prime}(t)+2 y=8\) (b) \(y^{\prime \prime \prime}(t)+y^{\prime \prime}(t)+3 y^{\prime}(t)=1\) (c) \(3 y^{\prime \prime \prime}(t)+9 y^{\prime \prime}(t)=1\) \((d) y^{(4)}(t)+y^{\prime \prime}(t)=4\)

Short answer

(a) 4, (b) 1/10, (c) 1/9t, (d) 4.

Step-by-step solution

Understand the Problem

We need to find the particular integral of each given differential equation. The particular integral is a solution of the non-homogeneous differential equation.

Solve Equation (a)

The given equation is \( y''(t) + 2y'(t) + y'(t) + 2y = 8 \). The homogeneous part is \( y'' + 2y' + 2y' + y = 0 \), and using the method of undetermined coefficients, assume a particular solution of the form \( y_p = C \). Substitute into the equation to solve for \( C \). Since the non-homogeneous term is constant, the particular solution is simply a constant \( y_p = 8/2 = 4 \). Thus, \( y_p = 4 \).

Solve Equation (b)

The given equation is \( y''' + y'' + 3y' = 1 \). We can try a particular solution of \( y_p = At \) since the non-homogeneous term is a constant, differentiate and solve for \( A \) by substituting back into the differential equation. You will find \( A = 0 \) does not satisfy this because of errors in considering the characteristic equation which should be applied for complementary solutions. Alternatively, try \( y_p = C \) and proceed finding \( C \). Substitute back to find \( y_p = 1/10 \).

Solve Equation (c)

The given equation is \( 3y''' + 9y'' = 1 \). Assume a particular solution \( y_p = A \). In such a case, substitute and solve for \( 3*0 + 9*0 = 1 \) simply holds for no direct A solution. Attempt \( y_p = Ct \) as it applies to linear components where solved \( Ct \) then gives no restriction. Solve mainly using a matching method since the derived function form matches a non-zero scenario leading to (Ct^3/54 + Ct/9). Particular solution is derived for A leading me up as \( y_p = 1/9t \).

Solve Equation (d)

The given equation is \( y^{(4)} + y'' = 4 \). Assume the form \( y_p = Ct^0 \), substitute and differentiate to satisfy the given non-homogeneous equation. Through differentials leave and solving \( C \) value of linear adjustment. Identify \( C \) via recursive differential nature capping into given non-components i.e., \( C = 1 \) innovates. Finalized as \( y_p = 4 \), giving non-homogeneous particular scenarios envisaged within return contribution.

Key concepts

Non-Homogeneous Differential Equations

Non-homogeneous differential equations are equations that include a non-zero function on the right side of the equation. These are different from homogeneous equations, where the right side is equal to zero. Understanding whether a differential equation is homogeneous or non-homogeneous is the first step to finding its solution.

In these types of equations, you are looking to find the "particular integral," a solution that incorporates the non-zero element. The equation often takes the form:

• Homogeneous Part: Solves the left part of the equation.
• Particular Solution: Accounts for the non-zero part on the right side.

By breaking down the equation into these components, you make it more manageable and easier to solve. This concept is crucial in various scientific fields, including engineering and physics.

Undetermined Coefficients Method

The Method of Undetermined Coefficients is a straightforward technique to find a particular solution for a non-homogeneous differential equation. This method works best when the non-homogeneous part of the equation is a polynomial, an exponential function, a sine or cosine, or a combination of these.

The process involves guessing the form of the particular solution based on the type of non-homogeneous term. For example:

• If the term is a constant, guess a constant solution.
• If the term is linear, guess a linear form solution.
• If the term is exponential, guess an exponential solution of the same form.

Once you have your guess, differentiate it accordingly and substitute it back into the original differential equation. Solving the equation after substitution will give you the "undetermined coefficients," which are the exact values or functions needed to solve the particular integral. This method saves time in mathematical problem solving by avoiding more complex calculations.

Mathematical Problem Solving

Problem-solving in mathematics, especially with differential equations, is a systematic process. It's critical to carefully read and understand the problem before attempting to solve it.

Here are some steps to take:

• Identify what type of differential equation you are dealing with: homogeneous or non-homogeneous.
• Recognize the form of the non-homogeneous component to determine the appropriate method for finding the particular integral.
• Use methods like undetermined coefficients or variation of parameters to find the solution.

Mistakes can often happen when not paying proper attention to each component of the problem, so always double-check your work. Solving differential equations is like solving a puzzle, where each piece must connect seamlessly with the next.

Differential Equations

Differential equations are mathematical equations that involve the derivatives of a function or functions. These equations define relationships involving rates of change, crucial for modeling real-world phenomena.

The main types are:

• Ordinary Differential Equations (ODEs): Involving functions of one independent variable.
• Partial Differential Equations (PDEs): Involving functions of multiple independent variables.

Understanding these concepts is important in fields like physics, where they describe physical processes, or in engineering for system modeling. Mastery of differential equations provides the ability to describe the dynamics of various systems and interpret how they evolve over time.

Mathematical Economics

Mathematical Economics involves using mathematical methods to represent economic theories and analyze economic problems. Differential equations form the backbone of many economic models to describe dynamic systems.

For instance, they can model how an economy evolves over time, capturing key relationships like growth rates, capital accumulation, or consumer behavior.

• They allow economists to predict future behaviors based on a set of initial conditions.
• Help to understand complex economic concepts and represent them in a form amenable to policy analysis or strategic planning.

By applying differential equations, mathematical economics opens avenues for creating more precise and testable economic theories, bridging gaps between abstract theory and real-world data.