Problem 28 - Calculus for Biology and Medicine

Question

Transform the second-order differential equation $$ \frac{d^{2} x}{d t^{2}}-2 \frac{d x}{d t}=3 x $$ into a system of first-order differential equations.

Step-by-Step Solution

Answer

The system is \( \frac{dy_1}{dt} = y_2 \) and \( \frac{dy_2}{dt} = 2y_2 + 3y_1 \).

Step 1: Introduction of Variables

Introduce new variables to transform the second-order equation into a system of first-order equations. Let \( y_1 = x \) and \( y_2 = \frac{dx}{dt} \). This establishes a relationship between \( x \) and its first derivative \( \frac{dx}{dt} \).

Step 2: Rewrite Derivatives

Express the derivatives in terms of the new variables. We know \( y_1 = x \), thus \( \frac{dy_1}{dt} = \frac{dx}{dt} = y_2 \). This gives us the first equation of the system: \( \frac{dy_1}{dt} = y_2 \).

Step 3: Transform Second-Order Derivative

The second derivative \( \frac{d^2 x}{dt^2} \) can be written as \( \frac{d}{dt}(\frac{dx}{dt}) = \frac{dy_2}{dt} \). Substitute this into the original equation: \( \frac{dy_2}{dt} - 2 y_2 = 3 y_1 \).

Step 4: Final System of Equations

Combine both derived first-order equations into a system:- \( \frac{dy_1}{dt} = y_2 \)- \( \frac{dy_2}{dt} = 2y_2 + 3y_1 \).This represents the system of first-order differential equations.

Key Concepts

Second-Order Differential Equations

Second-order differential equations involve derivatives up to the second degree. These equations take the general form \( \frac{d^2x}{dt^2} = f(t, x, \frac{dx}{dt}) \).Their solutions reveal information about the behavior and dynamics of systems that change over time.
Understanding and solving these equations is crucial as they often appear in modeling physical processes such as motion in classical mechanics, electrical circuits, and other natural phenomena.

• The second derivative, \( \frac{d^2x}{dt^2} \), describes how the rate of change of a quantity itself changes over time.
• This means it provides insight into acceleration, curvature, or more complex forms of behavior changes.

To solve these equations, we often employ techniques like variable substitution or integrating factors, depending on their complexity and form. However, they can often be transformed into a more manageable set of equations like first-order differential equations.

Differential Equation Systems

Differential equation systems refer to a set of related differential equations that describe a complex system or process.
These systems might involve multiple interdependent variables and their derivatives. In our exercise, we started with a second-order equation and transformed it into a system of first-order equations.

• Converting higher-order equations into a system of first-order equations helps simplify analysis and use existing methods for solving them.
• Each equation in the system usually corresponds to a specific variable and its derivative, providing a clear picture of how changes unfold over time.

By studying the system of equations, we can gain insights into the dynamic behavior of multi-variable systems.
This is particularly useful in engineering, physics, and economics to model processes that evolve with interdependent variables.

Variable Substitution

Variable substitution is a powerful method used to simplify and solve differential equations by altering the dependent variables.
This technique involves introducing new variables to replace derivatives or combine terms, transforming the equation's complexity.
In the original exercise, the substitution was as follows:

• Let \( y_1 = x \) and \( y_2 = \frac{dx}{dt} \).

This substitution allowed us to convert the second-order differential equation into a set of first-order equations:

• \( \frac{dy_1}{dt} = y_2 \)
• \( \frac{dy_2}{dt} = 2y_2 + 3y_1 \)

By solving this new system of equations, we can find solutions that may be more difficult to achieve with the original, more complex form.
Variable substitution not only simplifies the mathematical process but also unveils insights through different perspectives within the same problem context.