Problem 35 - Calculus for Biology and Medicine

Question

Transform the second-order differential equation $$ \frac{d^{2} x}{d t^{2}}=-\frac{1}{2} x $$ into a system of first-order differential equations.

Step-by-Step Solution

Answer

Introduce \( v = \frac{dx}{dt} \), then the system is \( \frac{dx}{dt} = v \) and \( \frac{dv}{dt} = -\frac{1}{2}x \).

Step 1: Define New Variables

Let's introduce a new variable to transform the second-order equation into a system of first-order differential equations. Define the new variable as follows: let \( v = \frac{dx}{dt} \). Now we have a system where \( \frac{dv}{dt} = -\frac{1}{2}x \) based on the original equation.

Step 2: Write System of Equations

The original second-order differential equation \( \frac{d^{2} x}{d t^{2}} = -\frac{1}{2} x \) can now be expressed using the new variable \( v \) as a system of two first-order equations:1. \( \frac{dx}{dt} = v \)2. \( \frac{dv}{dt} = -\frac{1}{2} x \)

Step 3: Verify the System

Verify that the system correctly represents the original second-order differential equation. The equation \( \frac{dx}{dt} = v \) defines \( v \) as the derivative of \( x \) with respect to \( t \). Substituting \( v \) into \( \frac{dv}{dt} = -\frac{1}{2} x \) ensures that this system is equivalent to the original second-order equation.

Key Concepts

Second-Order Differential Equations

Second-order differential equations involve the second derivative of a function. This simply means they are concerned with the rate of change of the rate of change, or how fast the velocity is changing in the context of motion. In the original exercise, the equation is \[\frac{d^{2} x}{d t^{2}}=-\frac{1}{2} x\]This is a linear differential equation, and our goal is to transform it into something more manageable. Linear second-order differential equations are particularly important because they often model physical phenomena such as oscillations and wave motions.
To simplify solving, it's useful to transform these equations into a system of first-order differential equations. This can be done through a change of variables, dealing with each derivative separately and helping simplify the problem.
Let's move to the system of equations part to see how we can break it down further.

System of Equations

A system of equations consists of multiple equations that are solved together. This technique is very helpful in various fields, from mathematics to engineering. For differential equations, creating a system can simplify solving by reducing the complexity of higher-order derivatives.
By breaking down a second-order differential equation into a system of first-order equations, we handle simpler parts. In our example, we introduced a variable \( v \) which represents the first derivative \( (\frac{dx}{dt}) \). Now the original complex equation translates into two simpler first-order equations:

• \( \frac{dx}{dt} = v \)
• \( \frac{dv}{dt} = -\frac{1}{2} x \)

We essentially create a coupled system where one equation defines \( v \) based on \( x \), and the other shows the rate of change of \( x \). This decoupling makes it much easier to analyze the behavior of the system.
Understanding how to form and solve these systems is crucial in fields like physics and economics, where dynamic systems modeling is prevalent.

Variable Substitution

Variable substitution is a mathematical technique used to simplify complex problems by introducing new variables. This is especially useful in differential equations to transform them into a more workable form.
In our exercise, we used substitution to introduce a new variable \( v \), which simplifies our second-order differential equation. The substitution process acts like a task delegation, where each variable manages one of the derivatives:

• \( v = \frac{dx}{dt} \) reduces the equation's order by representing the first derivative.
• \( \frac{dv}{dt} = -\frac{1}{2} x \) simplifies the integration of the system, breaking down complex dynamics into easier computations.

This technique is common when faced with higher-order differential equations, as it helps restructure the equations, enhancing the ability to solve them analytically or numerically. Getting comfortable with choosing the right substitutions is a valuable skill in mathematics, as it opens up opportunities to tackle otherwise intractable problems.