Problem 14 - A First Course in Differential Equations with Modeling Applications

Question

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}+\frac{d y}{d t} \quad=e^{t}\\\ &-\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y=0 \end{aligned}$$

Step-by-Step Solution

Answer

Solve \( \frac{d y}{d t} = e^{t} - \frac{d x}{d t} \) and substitute into the second equation to find \( x(t) \) and \( y(t) \).

Step 1: Express Dy/Dt in Terms of Dx/Dt

From the first equation, \( \frac{d x}{d t} + \frac{d y}{d t} = e^{t} \), we can express \( \frac{d y}{d t} \) as \( \frac{d y}{d t} = e^{t} - \frac{d x}{d t} \). This equation will be used to substitute \( \frac{d y}{d t} \) in the second equation.

Step 2: Substitute \( \frac{d y}{d t} \) into the Second Equation

Substitute \( \frac{d y}{d t} = e^{t} - \frac{d x}{d t} \) into the second equation \( -\frac{d^{2} x}{d t^{2}} + \frac{d x}{d t} + x + y = 0 \). This gives us one equation in terms of \( x \), \( \frac{d x}{d t} \), and \( \frac{d^{2} x}{d t^{2}} \).

Step 3: Differentiate the First Equation

Differentiate the first equation with respect to \( t \) to get \( \frac{d}{dt}(\frac{d x}{d t} + \frac{d y}{d t}) = \frac{d}{dt}(e^{t}) \). This simplifies to \( \frac{d^{2} x}{d t^{2}} + \frac{d^{2} y}{d t^{2}} = e^{t} \).

Step 4: Solve the Second Order Differential Equation

Using the results from the previous steps, solve the system of differential equations derived for \( \frac{d^{2} x}{d t^{2}} \) and \( \frac{d x}{d t} \) to find the particular and complementary solution for \( x(t) \).

Step 5: Back-Substitution to Find y(t)

Use the expression for \( \frac{d y}{d t} = e^{t} - \frac{d x}{d t} \) and the solution for \( x(t) \) to find \( y(t) \). Substitute back if necessary to verify the solution satisfies both original equations.

Key Concepts

System of Differential Equations

A system of differential equations involves multiple equations with multiple unknown functions and their derivatives. These systems can model various real-world processes, such as biological systems or mechanical systems, where different variables interact with each other dynamically. The system of differential equations we work on here consists of two equations:

• \( \frac{dx}{dt} + \frac{dy}{dt} = e^t \)
• \(-\frac{d^2x}{dt^2} + \frac{dx}{dt} + x + y = 0 \)

These equations describe the relationship between function \( x(t) \) and \( y(t) \) and their rates of change. Solving such a system means finding the functions \( x(t) \) and \( y(t) \) that satisfy both equations simultaneously. We often need to manipulate these equations using algebraic techniques to reduce the system to simpler or more familiar forms which can then be solved.

Order Reduction

Order reduction is a crucial technique used to simplify differential equations by reducing their order, which is the highest derivative present in the equation. In the given system, the presence of second derivatives (for \( x(t) \)) indicates a second-order differential equation.

By applying the principle of order reduction, the problem-solving process becomes more manageable. We achieve this by expressing one of the variables or its derivative in terms of another via substitution. In this exercise, a reduction in order was performed by differentiating the first equation and substituting the expressions into the second, turning the original second-order equation into a more tractable form. Reducing the order of differential equations is essential because it often transforms the equations into the ones we are used to solving, like first-order linear differential equations, which have well-established solution methods.

Elimination Method

The elimination method is a powerful tool for solving systems of equations, whether they are algebraic or differential. In this method, we aim to eliminate one of the variables or terms to simplify the system into a single equation that can be solved more straightforwardly. This approach involves these general steps:

• Manipulating the equations to express one variable or derivative explicitly in terms of the other.
• Substituting the expression into the remaining equations.
• Simplifying the system until a single differential equation is obtained.

In the example exercise, we start by expressing \( \frac{dy}{dt} \) from the first equation and use this expression to substitute into the second equation. It allows us to handle one equation dependent entirely on \( x \) and its derivatives. Finally, we simplify this into a form where known techniques can be applied to find a solution. The elimination method streamlines the process of solving systems, effectively reducing the complexity involved.

Particular and Complementary Solutions

In solving differential equations, particularly linear ones, it's crucial to find both the particular and complementary solutions to form the complete solution. Let's explore these terms:

• The **complementary solution** is associated with the homogeneous part of the equation, which is where the right-hand side is zero. It's typically found using methods like characteristic equations when dealing with constant coefficients.
• The **particular solution** addresses the non-homogeneity of the equation. It ensures that any external forces or inputs into the system (represented by the non-zero right-hand side) are accounted for in the solution.

Together, the combination of these solutions gives the general solution: \[ x(t) = x_c(t) + x_p(t) \]In the given exercise, you derive both solutions to solve for \( x(t) \), ensuring that reflections of both the system's natural behavior and the effects of external inputs are included. Once \( x(t) \) is fully solved, you can substitute back to find the complete solution for \( y(t) \), ensuring that both functions \( x(t) \) and \( y(t) \) satisfy the initial set of differential equations.