Question

Transform the given equation into a system of first order equations. \(u^{\prime \prime}+0.5 u^{\prime}+2 u=0\)

Short answer

Question: Transform the given second-order differential equation into a system of first-order differential equations: \(u''+0.5u'+2u=0\). Answer: The equivalent system of first-order differential equations is given by: \begin{cases} u' = v \\ v' = -0.5v - 2u \end{cases}

Step-by-step solution

Introduce a new variable

Let \(v=u'\). Now, we have \(u''=v'\), and we will substitute this new variable into the given equation.

Rewrite the equation using the new variables

Substituting the new variables into the given equation, we get \(v'+0.5v+2u=0\).

Express the system of first-order equations

Now, we already have \(u'=v\), and from Step 2, we have derived \(v'+0.5v+2u=0\). Thus, the system of first-order differential equations is given by: \begin{cases} u' = v \\ v' = -0.5v - 2u \end{cases} This system of first-order equations is equivalent to the original second-order equation and can be solved using relevant methods for first-order equations.

Key concepts

Understanding Differential Equations

When we talk about differential equations, we are pointing to one of the most pivotal tools in applied mathematics, physics, and engineering. These equations enable us to describe how certain quantities change over time or space. Essentially, a differential equation links some function with its derivatives, expressing the rate at which this function is changing.

In the context of our exercise, the differential equation in question is a second-order equation because it involves the second derivative of the function, denoted as \(u''\). Each order of a differential equation corresponds to the degree of derivatives involved. Higher-order differential equations, like the one in the exercise, are commonly encountered in problems involving oscillations, electronic circuits, and many other systems.

To effectively tackle differential equations, one must thoroughly understand the concepts of derivatives, integration, and their role in expressing physical laws. Solving a differential equation often involves finding a function that fulfills the equation, which can represent a model of a real-life phenomenon.

The Role of Variable Substitution

A valuable technique used to solve higher-order differential equations is variable substitution. This approach simplifies complex equations, making them more manageable. The core idea is to replace higher-order derivatives with new variables, transforming the equation into a system of first-order differential equations.

In our sample exercise, we introduced the variable \(v=u'\), which represents the first derivative of \(u\). This simple yet clever substitution greatly simplifies the analysis and solution process. By redefining the components of the equation, we distill a higher-order equation into a more approachable system.

One of the primary advantages of variable substitution is that it unlocks the potential to apply specialized methods, suited for first-order equations, to a broader range of problems. An in-depth understanding of how and when to perform variable substitution can immensely benefit students as they delve into advanced topics in calculus and differential equations.

From Higher-Order Differential Equations to System Conversion

The transformation from a single higher-order differential equation to a system of first-order differential equations is a transformation that expands the scope of mathematical tools available for solving complex equations. Such a conversion is especially useful for numerical methods, which often require the use of systems of first-order equations.

In our tutorial's step-by-step solution, we have achieved this transformation. Starting with the original second-order equation \(u''+0.5 u'+2 u=0\), we utilized our new variable \(v\) and expressed the equation as a system of two first-order equations. Now the problem can be approached with methods like the Euler method, Runge-Kutta methods, or even qualitative analysis, which provide insights into the system's behavior without requiring a closed-form solution.

Such conversions not only demystify complex equations but also equip students with robust methodologies to approach a wide array of mathematical and real-world problems, demonstrating the interconnectedness of mathematical concepts and the importance of a strong foundation in differential equations.