Chapter 9: Problem 7
Find the Wronskian \(W\) of a set of three solutions of $$ y^{\prime \prime \prime}+2 x y^{\prime \prime}+e^{x} y^{\prime}-y=0, $$ given that \(W(0)=2\).
Short Answer
Expert verified
Answer: The Wronskian of the set of three solutions of the given third-order linear ODE is: \(W(x) = 2 \exp(1-e^x)\).
Step by step solution
01
State the definition of the Wronskian for a set of three solutions
For a set of three solutions \(y_1(x)\), \(y_2(x)\), and \(y_3(x)\) of a third-order linear ordinary differential equation, the Wronskian \(W\) is defined as the determinant of the following matrix:
$$
W(y_1, y_2, y_3) = \begin{vmatrix}
y_1 & y_2 & y_3 \\
y_1' & y_2' & y_3' \\
y_1'' & y_2'' & y_3'' \\
\end{vmatrix}.
$$
02
Write down the given ODE and the initial condition
The given third-order linear ODE is
$$
y^{\prime \prime \prime}+2 x y^{\prime \prime}+e^{x} y^{\prime}-y=0,
$$
and the initial condition for the Wronskian is \(W(0) = 2\).
03
Apply Abel's theorem
Abel's theorem states that the Wronskian of n linearly independent solutions of an n-th order linear homogeneous ODE is given by:
$$
W(x) = W(x_0) \exp \left( -\int^{x}_{x_0} p_1(s) ds \right),
$$
where \(p_1(x)\) is the coefficient of the first derivative term in the ODE.
04
Identify the coefficient of the first derivative term
In our given ODE,
$$
y^{\prime \prime \prime}+2 x y^{\prime \prime}+e^{x} y^{\prime}-y=0,
$$
the coefficient of the first derivative term \(y'\) is \(p_1(x) = e^x\).
05
Calculate the integral of the coefficient
Find the integral of the coefficient of the first derivative term:
$$
\int p_1(s) ds = \int e^s ds = e^s + C,
$$
where \(C\) is a constant of integration. In this case, the constant will not affect our calculation since we will be using a definite integral, so we can disregard the constant.
06
Apply Abel's theorem with the initial condition
Applying Abel's theorem using the initial condition \(W(0) = 2\), we have:
$$
W(x) = W(0) \exp \left(-\int^x_0 e^s ds\right) = 2 \exp \left(-\int^x_0 e^s ds\right).
$$
Now, integrate:
$$
W(x) = 2 \exp \left(-(e^x - e^0)\right) = 2 \exp(1-e^x).
$$
07
State the final answer
The Wronskian of the set of three solutions of the given third-order linear ODE is:
$$
W(x) = 2 \exp(1-e^x).
$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Differential Equations
Differential equations are mathematical equations that relate some function with its derivatives. They describe a wide range of phenomena in nature and are fundamental in fields like physics, engineering, biology, economics, and beyond. A typical differential equation defines the rate at which a process happens, often leading to a formula for the process itself.
For instance, if you're studying the rate of growth of a population of animals, you might use a differential equation to express the population growth in relation to various factors like available resources, existing population, and time.
Differential equations can be categorized in several ways, with two primary classifications being ordinary differential equations (ODEs), which involve a single independent variable, and partial differential equations (PDEs), involving multiple independent variables.
For instance, if you're studying the rate of growth of a population of animals, you might use a differential equation to express the population growth in relation to various factors like available resources, existing population, and time.
Differential equations can be categorized in several ways, with two primary classifications being ordinary differential equations (ODEs), which involve a single independent variable, and partial differential equations (PDEs), involving multiple independent variables.
Linear Homogeneous ODE
A linear homogeneous ordinary differential equation (ODE) is a type of differential equation that has several unique and convenient properties. The 'linear' part of the name indicates that the equation is a linear function of the unknown function and its derivatives, while 'homogeneous' implies that all the terms are multiples of the unknown function or its derivatives without any constant or otherwise 'non-homogenous' terms.
To put it simply, in a linear homogeneous ODE, each term involves either the unknown function or one of its derivatives, and the sum of all these terms equals zero. This property guarantees that if you have several solutions to the equation, any linear combination of these solutions is also a solution - this is known as the principle of superposition.
Understanding the structure of a linear homogeneous ODE is crucial as it helps to determine the methods appropriate for finding its solutions, which are vital for accurately describing the scientific or engineering problems modeled by the equations.
To put it simply, in a linear homogeneous ODE, each term involves either the unknown function or one of its derivatives, and the sum of all these terms equals zero. This property guarantees that if you have several solutions to the equation, any linear combination of these solutions is also a solution - this is known as the principle of superposition.
Understanding the structure of a linear homogeneous ODE is crucial as it helps to determine the methods appropriate for finding its solutions, which are vital for accurately describing the scientific or engineering problems modeled by the equations.
Abel's Theorem
Abel's theorem is a powerful tool in the study of differential equations. Named after the Norwegian mathematician Niels Henrik Abel, this theorem is concerned with the Wronskian of linearly independent solutions to a linear differential equation. The Wronskian is a determinant associated with a set of functions, which in this context, is used to determine if a set of solutions to an ODE is linearly independent.
According to Abel's theorem, for an n-th order linear homogeneous ODE, the Wronskian of n linearly independent solutions does not vanish, and it's possible to express it in terms of an exponential function that involves an integral of the coefficient of the first derivative from the ODE. This relationship is immensely useful because it allows you to understand how the set of solutions varies with the independent variable, and it provides insight into the behavior and properties of the solutions to the ODE.
According to Abel's theorem, for an n-th order linear homogeneous ODE, the Wronskian of n linearly independent solutions does not vanish, and it's possible to express it in terms of an exponential function that involves an integral of the coefficient of the first derivative from the ODE. This relationship is immensely useful because it allows you to understand how the set of solutions varies with the independent variable, and it provides insight into the behavior and properties of the solutions to the ODE.
Initial Conditions
Initial conditions are values specified for the function and its derivatives at a particular point, and they are essential for solving ordinary differential equations uniquely. These conditions are critical since most ODEs will have an infinite number of possible solutions without them.
Specifying an initial condition allows you to pin down which particular solution is relevant for the physical or mathematical problem you're considering. For a third-order ODE, typically three initial conditions are needed to determine a unique solution: the function's value at a point, and its first and second derivatives at the same point.
In practical terms, the initial condition often represents the state of the system at the start of an observation or experiment. For example, knowing the initial position and velocity of a projectile would be essential in predicting its subsequent motion under the influence of gravity.
Specifying an initial condition allows you to pin down which particular solution is relevant for the physical or mathematical problem you're considering. For a third-order ODE, typically three initial conditions are needed to determine a unique solution: the function's value at a point, and its first and second derivatives at the same point.
In practical terms, the initial condition often represents the state of the system at the start of an observation or experiment. For example, knowing the initial position and velocity of a projectile would be essential in predicting its subsequent motion under the influence of gravity.