Question

The given differential equation has a fundamental set of solutions whose Wronskian \(W(t)\) is such that \(W(0)=1\). What is \(W(4) ?\) \(y^{\prime \prime \prime}+\frac{t}{2} y^{\prime}+y=0\)

Short answer

Answer: The value of the Wronskian at \(t=4\) is \(W(4) = e^{-4}\).

Step-by-step solution

Write the given differential equation in standard form

Before using Abel's Identity, we should rewrite the given differential equation in standard form, which has the form: \(y^{\prime \prime \prime} + p(t) y^{\prime} + q(t) y = 0.\) Comparing the given differential equation to the standard form, we can see it is already in standard form, with \(p(t) = \frac{t}{2}\) and \(q(t) = 1\).

Find the expression for the Wronskian using Abel's Identity

Abel's Identity states that for an \(n\)-th order linear homogeneous differential equation, the Wronskian \(W(t)\) is given by: \(W(t) = ce^{-\int p(t)dt}\), where \(c\) is an integration constant. In our case, \(n=3\), and \(p(t) = \frac{t}{2}\), so we need to compute the integral: \(-\int p(t)dt = -\int \frac{t}{2} dt\).

Compute the integral

Compute the integral in the expression for the Wronskian: \(-\int \frac{t}{2} dt = -\frac{1}{2} \int t dt = -\frac{1}{2} \cdot \frac{1}{2} t^2 + C.\)

Find the integration constant

Since \(W(0)=1\), plug in the value of \(t=0\) into the expression for the Wronskian and solve for the constant \(c\): \(1 = ce^{-\frac{1}{2}\cdot \frac{1}{2} \cdot 0^2}\). Since the exponent is zero, the expression simplifies to: \(1 = c \cdot 1 \Rightarrow c = 1\).

Find the expression for the Wronskian with the constant

Now that we have the constant \(c\), we can write the full expression for the Wronskian: \(W(t) = e^{-\frac{1}{2}\cdot \frac{1}{2} t^2}\).

Evaluate the Wronskian at \(t=4\)

Now, we simply need to plug in the value of \(t=4\) to find the value of the Wronskian at \(t=4\): \(W(4) = e^{-\frac{1}{2}\cdot \frac{1}{2} \cdot 4^2}\), \(W(4) = e^{-\frac{1}{2}\cdot 8}\), \(W(4) = e^{-4}\). So, the value of the Wronskian at \(t=4\) is \(W(4) = e^{-4}\).

Key concepts

Abel's Identity

Abel's Identity is a powerful tool in differential equation analysis, providing an elegant method to find the Wronskian, which is a type of determinant used to assess the linear independence of a set of functions or solutions to a differential equation.

More specifically, for an n-th order linear homogeneous differential equation like the one in the exercise \(y^{\prime \prime \prime} + p(t) y^{\prime} + q(t) y = 0\), Abel's Identity tells us that the Wronskian, \(W(t)\), relates directly to the coefficient function \(p(t)\) of the first derivative term. Using Abel's Identity, the Wronskian is expressed as follows:
\[W(t) = ce^{-\int p(t)dt}\]
where \(c\) is an integration constant that can be determined from initial conditions, and the integral involves an antiderivative of the coefficient function \(p(t)\).

In practice, applying Abel’s Identity involves computing the integral of \(p(t)\) and adjusting for initial conditions, ultimately providing the Wronskian without solving the entire differential equation, which could be exceedingly complex.

Linear Homogeneous Differential Equation

A linear homogeneous differential equation is a special class of differential equations characterized by the fact that every term is either a derivative of the unknown function or the function itself, multiplied by a function of the independent variable.

An n-th order linear homogeneous differential equation has the general form:
\[a_n(t)y^{(n)} + a_{n-1}(t)y^{(n-1)} + \ldots + a_1(t)y^{\prime} + a_0(t)y = 0\]
Where \(a_n(t), \ldots, a_0(t)\) are functions of the independent variable \(t\) only. The 'homogeneous' part of the name refers to the fact that there is no term independent of the unknown function \(y\), such as a constant or a function of \(t\) alone.

One of the key features of homogeneous equations is that if \(y_1\) and \(y_2\) are solutions, then any linear combination \(c_1y_1 + c_2y_2\) is also a solution, where \(c_1\) and \(c_2\) are constants. This principle embodies the superposition property, which is crucial in solving these types of differential equations.

Integration Constant

The integration constant is a pivotal component in calculus, particularly when dealing with indefinite integrals, or antiderivatives. When we perform the indefinite integration of a function, we include an unknown constant, \(C\), to account for all the possible antiderivatives.

This constant represents the infinite number of vertical translations a curve might undergo and still retain the same derivative. In other words, the general antiderivative of a function \(f(t)\) with respect to \(t\) is written as:
\[\int f(t)dt = F(t) + C\]
where \(F(t)\) is an antiderivative of \(f(t)\) and \(C\) is the integration constant.

In the context of differential equations, the integration constant takes on a more nuanced role. It helps in determining the particular solution to the differential equation that satisfies given initial conditions. As seen in our exercise, the initial value \(W(0)=1\) allowed us to find the specific value for the integration constant \(c\) in Abel's Identity, leading to the complete expression for the Wronskian. This illustrates the integration constant's part in anchoring abstract solutions to concrete scenarios.