Question

Use Abel’s formula (Problem 20) to find the Wronskian of a fundamental set of solutions of the given differential equation. $$ t^{2} y^{\mathrm{iv}}+t y^{\prime \prime \prime}+y^{\prime \prime}-4 y=0 $$

Short answer

Answer: The general expression for the Wronskian is \(W(t) = W_0t^{1-\frac{1}{t}}\).

Step-by-step solution

Calculate \(\gamma\)

We'll first find the value of \(\gamma\). In this case, \(c_0 = t^2\) and \(c_1 = t\), so: $$ \gamma = \frac{c_1}{c_0} = \frac{t}{t^2} = \frac{1}{t} $$

Apply Abel's Formula

Now we'll use Abel's formula to find the Wronskian \(W(t)\). We know that \(W(t) = W_0t^{1-\gamma}\). Let's substitute the value of \(\gamma\) in the formula and find the Wronskian: $$ W(t) = W_0t^{1-\frac{1}{t}} $$ In this equation, W_0 represents the Wronskian at a specific initial value, usually denoted as W(a). It indicates that the Wronskian formula is true for any value given the initial conditions. Unfortunately, we don't have information about the particular initial conditions for this problem. So without specific initial conditions being given, we won't be able to calculate the value of \(W_0\), which means we won't be able to find the Wronskian for a specific set of initial conditions. However, we have found the general expression for the Wronskian of the differential equation in terms of \(W_0\) and \(t\): $$ W(t) = W_0t^{1-\frac{1}{t}} $$

Key concepts

Abel's Formula

Abel's Formula is a useful tool in the study of differential equations, especially when dealing with linear equations. It is named after the mathematician Niels Henrik Abel. The formula provides a way to find the Wronskian of a set of solutions without having to solve the differential equation directly. This can save time and effort in many scenarios.

• The Wronskian, denoted as \(W(t)\), is a determinant used to determine whether a set of solutions is linearly independent. If \(W(t) eq 0\), the solutions are linearly independent.
• Abel's formula states that \(W(t) = W_0 t^{1-rac{1}{t}}\) for the differential equation given, where \(W_0\) is a constant based on the initial conditions of the problem.
• It is important to note that Abel's formula is particularly suited for second-order linear differential equations but can also be extended to higher orders with modifications.

Understanding Abel's formula requires a fundamental grasp of linear algebra concepts, particularly determinants, as they relate to systems of equations.

Differential Equation

Differential equations are mathematical equations that relate a function with its derivatives. They play a critical role in modeling real-world phenomena in physics, engineering, economics, and beyond. The equation given in the exercise represents a higher-order differential equation:

• The equation is \(t^2 y^{\mathrm{iv}} + t y^{\prime \prime \prime} + y^{\prime \prime} - 4y = 0\), involving derivatives of the unknown function \(y(t)\) up to the fourth order.
• Such equations often describe systems with dynamic behavior, such as oscillations or wave patterns.
• Solving higher-order differential equations can be complex, and methods like Abel's Formula can simplify certain steps.Understanding the structure and solutions of these equations provides insights into the potential behavior of the system they model.

By learning how to manipulate and solve differential equations, students can predict how systems evolve over time.

Fundamental Set of Solutions

A fundamental set of solutions to a differential equation consists of solutions that form a basis for the solution space. In other words, any solution of the differential equation can be expressed as a linear combination of the elements in the fundamental set.

• For the given differential equation, if you have \(n\) linearly independent solutions, those solutions make up a fundamental set.
• A fundamental set is crucial because it allows for the complete general solution of the differential equation to be written as \( y(t) = C_1y_1(t) + C_2y_2(t) + ... + C_ny_n(t) \), where \(C_1, C_2, ..., C_n\) are constants determined by initial or boundary conditions.
• The independence of the solutions, determined by the Wronskian, ensures that the solutions can span the entire solution space, providing flexibility in fitting to any set of initial conditions.

Having a solid understanding of fundamental sets of solutions is essential for effectively utilizing differential equations in practical applications.

Initial Conditions

Initial conditions are essential details specified at the start of a problem that allow us to find particular solutions to differential equations out of the general solution. These conditions are used to calculate the constants in the general solution.

• Initial conditions provide information about the state of the system at a specific time, often \( t = t_0 \).
• For instance, given the general solution of a differential equation, applying initial conditions such as \( y(t_0) = y_0 \) or \( y'(t_0) = y'_0 \) will allow us to solve for the constants \(C_1, C_2, ..., C_n\).
• In the context of the step-by-step solution, the Wronskian \(W_0\) is determined by specific initial conditions, which were not provided, leaving \(W(t)\) in terms of an arbitrary constant.

Without initial conditions, a complete specific solution cannot be determined, emphasizing their importance in solving real-world problems using differential equations.