Question

Transform the given equation into a system of first order equations. \(t^{2} u^{\prime \prime}+t u^{\prime}+\left(t^{2}-0.25\right) u=0\)

Short answer

Question: Transform the given second-order differential equation into a system of first-order equations: \(t^2 u''(t) + tu'(t) + (t^2 - 0.25)u(t) = 0\). Answer: The given second-order differential equation can be transformed into a system of two first-order equations as follows: (1) \(u'(t) = v(t)\), (2) \(t^2 v'(t) + t v(t) + (t^2 - 0.25)u(t) = 0\).

Step-by-step solution

Introduce a new dependent variable for the first derivative of u

Let \(v(t) = u'(t)\). This new function represents the first derivative of \(u\).

Write the given equation in terms of the new variable

Substitute \(v\) for \(u'(t)\) and \(v'(t)\) for \(u''(t)\) in the given equation: \(t^2 v'(t) + t v(t) + (t^2 - 0.25)u(t) = 0\).

Separate the equation into a system of first-order equations

Write the original equation as two first-order equations with respect to \(u\) and \(v\): (1) \(u'(t) = v(t)\), (2) \(t^2 v'(t) + t v(t) + (t^2 - 0.25)u(t) = 0\), which represents a system of two first-order differential equations involving \(u(t)\) and \(v(t)\).

Key concepts

Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives. These equations often model dynamic processes in physics, engineering, biology, and finance, among other fields. For example, they can describe how a physical system changes over time or space. The key to understanding a differential equation is knowing that the solution is not a single number, but a function or a set of functions that satisfy the given relationship between the variable and its derivative.

In the context of our exercise, the given equation involves a second derivative, which makes it a second-order differential equation. It's important for students to recognize that a higher-order differential equation can be transformed into a system of first-order differential equations, making the system easier to solve with standard methods.

Boundary Value Problem

Boundary value problems are a type of differential equation problem where we are given certain conditions, known as 'boundary conditions', that the solution must satisfy at the boundaries of the domain. These domains could be spatial boundaries for physical problems or time intervals for temporal problems.

While boundary conditions were not specified in our exercise, it's useful to note that solving most real-world problems requires taking these conditions into account. Boundary conditions help us determine the specific solution from the infinite family of solutions that differential equations usually have. In a classroom or textbook scenario, understanding how to apply boundary conditions is essential for correctly solving these types of problems.

Transformation of Equations

Sometimes, solving a differential equation directly can be very challenging or practically impossible. To cope with this, we use a strategy known as the transformation of equations. This involves manipulations that convert complex equations into simpler forms or systems of equations that are more tractable.

In our exercise, the transformation involved introducing a new variable to convert the second-order differential equation into a system of first-order differential equations. This step simplifies the problem and opens up the possibility of using methods designed specifically for first-order systems, such as the method of characteristics or matrix exponentials for linear systems. Transformation techniques are invaluable tools in a mathematician's arsenal.

Ordinary Differential Equations

Ordinary differential equations (ODEs) are differential equations that contain one or more functions of one independent variable and its derivatives. They differ from partial differential equations (PDEs), which involve multiple independent variables. ODEs are termed 'ordinary' to indicate that they have a single variable, in contrast to the 'partial', which indicates multiple variables in PDEs.

The system we dealt with in this exercise consists of ODEs, as there is only one independent variable, which is time (t). Understanding the classification between ODEs and PDEs is crucial when selecting appropriate solution methods. For example, methods for solving first-order ODEs include separation of variables, integrating factors, and linear algebra techniques for systems of linear ODEs.