Problem 36 - Calculus for Biology and Medicine

Question

Transform the second-order differential equation $$ \frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}=2 x $$ into a system of first-order differential equations.

Step-by-Step Solution

Answer

The transformed system is: \(\frac{dy_1}{dt} = y_2\), \(\frac{dy_2}{dt} = 2y_1 - y_2\).

Step 1: Identify Variables and Equations

Given the second-order differential equation \(\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}=2 x\), we identify the variable \(x\) dependent on time \(t\). The goal is to express this equation as a system of first-order differential equations.

Step 2: Introduce a Substitution

Define a new variable \(y_1 = x\) and use this definition to facilitate reducing the order of the differential equation. This will allow us to express higher derivatives in terms of first-order derivatives.

Step 3: Define New Variables for Derivatives

Introduce a second variable \(y_2 = \frac{dx}{dt}\). This substitution helps express the second derivative \(\frac{d^2x}{dt^2}\) in terms of first-order derivatives with respect to \(y_1\) and \(y_2\).

Step 4: Express the System of Equations

From \(y_1 = x\), we have \(\frac{dy_1}{dt} = y_2\). The second-order differential equation \(\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}=2 x\) can be rewritten using \(y_1\) and \(y_2\) as \(\frac{dy_2}{dt} = 2y_1 - y_2\).

Step 5: System Formulation

We have now transformed the original second-order differential equation into the following system of first-order equations: \(\frac{dy_1}{dt} = y_2\) and \(\frac{dy_2}{dt} = 2y_1 - y_2\). This is the required transformation of equations.

Key Concepts

First-Order Differential Equations

First-order differential equations are equations that involve the first derivative of a function but no higher derivatives. They play a fundamental role in differential equations because they can describe a wide range of phenomena, from growth processes to decay processes, motion, and more.
Here's a clearer understanding:

• In a first-order differential equation, the highest derivative is the first derivative, typically written as \( \frac{dy}{dt} \), \( \frac{dx}{dt} \), or similar.
• The objective with these equations is often to find a function that satisfies the equation, which involves one of its derivatives.
• These equations can be linear or nonlinear depending on the function's terms.

In the context of the given problem, we transformed a second-order equation into two first-order differential equations. This is a common practice because first-order systems are often easier to analyze and solve.

Second-Order Differential Equations

Second-order differential equations involve the second derivative of a function, and they frequently appear in physics and engineering, particularly those involving acceleration or forces. Let's dive deeper into their nature:

• In these equations, you often see expressions like \( \frac{d^2y}{dt^2} \), which denotes the second derivative of \( y \) with respect to \( t \).
• They can describe systems with memory or inertia, such as the oscillation of a spring or an electric circuit.
• These equations may include terms of the first derivative and the function itself, adding to their complexity.

The provided equation \( \frac{d^{2} x}{d t^{2}} + \frac{d x}{d t} = 2x \) is a second-order linear differential equation. By transforming it into a system of first-order equations, we simplify the process of finding solutions.

System of Equations

A system of equations consists of two or more equations with multiple variables that are solved together. When handling differential equations, converting to a first-order system can simplify the analysis and computational solutions. Consider these important points about systems of equations:

• A system of first-order equations is essentially a set of equations containing first derivatives, often used to model dynamic systems in various fields.
• Each equation in the system is interlinked with others, allowing us to understand complex relationships between variables.
• The method of solving often involves techniques such as substitution, elimination, or matrix operations like determining eigenvalues and eigenvectors in linear algebra.

In the exercise, we took the initial second-order differential equation and created a system of equations: \( \frac{dy_1}{dt} = y_2 \) and \( \frac{dy_2}{dt} = 2y_1 - y_2 \). This system captures the original equation's behavior using first-order dynamics, a valuable technique for solving complex differential equations in various practical applications.