Question

If \(y_{1}\) and \(y_{2}\) are linearly independent solutions of \(t^{2} y^{\prime \prime}-2 y^{\prime}+(3+t) y=0\) and if \(W\left(y_{1}, y_{2}\right)(2)=3,\) find the value of \(W\left(y_{1}, y_{2}\right)(4)\)

Short answer

Based on the given second-order linear differential equation and Abel's theorem, we find that the value of the Wronskian W(y_1, y_2)(4) is equal to 3/4.

Step-by-step solution

Define the given differential equation and Wronskian

We are given the second-order linear differential equation: $$t^{2} y^{\prime \prime}-2 t y^{\prime}+(3+t) y=0.$$ And we know that \(y_1(t)\) and \(y_2(t)\) are linearly independent solutions of this equation. The Wronskian \(W(y_1, y_2)\) is defined as: $$W(y_1, y_2)(t) = y_1(t)y_2'(t) - y_1'(t)y_2(t).$$

Applying Abel's theorem to the differential equation

Abel's theorem states that for a second-order linear homogeneous differential equation, $$p(t)y'' + q(t)y' + r(t)y = 0, $$ the Wronskian can be found by: $$W(y_1, y_2)(t) = W(y_1, y_2)(t_0) \cdot e^{-\int_{t_0}^{t} \frac{q(s)}{p(s)} ds}.$$ In our case, \(p(t) = t^2\), \(q(t) = -2t\), and \(r(t) = 3+t\). So, the expression in the integral becomes, \(\frac{-2t}{t^2} = -\frac{2}{t}\).

Calculate the integral in Abel's theorem

We need to calculate the integral $$-\int_{t_0}^{t} \frac{2}{s} ds$$. This evaluates to: $$-2(\ln(t) - \ln(t_0)) = -2(\ln(\frac{t}{t_0}))$$.

Apply Abel's theorem to find the Wronskian for any t

Now, applying Abel's theorem with the calculated integral, we have: $$W(y_1, y_2)(t) = W(y_1, y_2)(t_0) \cdot e^{-2(\ln(\frac{t}{t_0}))}$$. We are given that \(W(y_1, y_2)(2) = 3\). Let's substitute \(t_0=2\) and \(t=4\) to find \(W(y_1, y_2)(4)\).

Find the value of the Wronskian at t = 4

Using \(t_0=2\), \(t=4\), and \(W(y_1, y_2)(2) = 3\), we get: $$W(y_1, y_2)(4) = 3 \cdot e^{-2(\ln(\frac{4}{2}))} = 3 \cdot e^{-2(\ln(2))} = 3 \cdot \frac{1}{4}.$$ Finally, we have: $$W(y_1, y_2)(4) = \frac{3}{4}.$$

Key concepts

Linear Independence

Linear independence is a fundamental concept in linear algebra and calculus. It refers to a set of functions or vectors where no member of the set can be written as a linear combination of others. In the context of differential equations, if two functions, say \(y_1(t)\) and \(y_2(t)\), are solutions to a differential equation, they are called linearly independent if the only way to weigh them, without canceling their effect entirely, is to use zero coefficients.

• This concept ensures that each function contributes its unique behavior to the solution of a differential equation.
• A basis of solutions to a differential equation, for instance, usually involves linearly independent functions to represent a wide range of phenomena.

Understanding linear independence helps in exploring a complete set of solutions and is crucial in fields like differential equations, where knowing all possible solutions is often essential for solving real-world problems.

Wronskian

The Wronskian is a determinant used to test for linear independence of functions. For two functions \(y_1(t)\) and \(y_2(t)\), their Wronskian \(W(y_1, y_2)(t)\) is defined as:\[W(y_1, y_2)(t) = y_1(t)y_2'(t) - y_1'(t)y_2(t).\]

• If the Wronskian is non-zero for some interval, the functions are linearly independent on that interval.
• It helps mathematicians verify if solutions to a differential equation are linearly independent, even if the solutions themselves aren't explicitly calculated.

In our problem, we computed the Wronskian at different points using Abel's theorem, demonstrating its useful role in working with linear systems and differential equations.

Abel's Theorem

Abel's Theorem is a crucial tool when dealing with second-order linear homogeneous differential equations. This theorem provides a way to calculate the Wronskian of two solutions without needing to know the solutions explicitly. For a second-order differential equation, given by:\[p(t) y'' + q(t) y' + r(t) y = 0,\]Abel's Theorem states that:\[W(y_1, y_2)(t) = W(y_1, y_2)(t_0) \cdot e^{-\int_{t_0}^{t} \frac{q(s)}{p(s)} ds}.\]

• It's particularly useful because it turns the problem of checking linear independence into an integration problem.
• This approach avoids the more complex task of solving the differential equation directly, leveraging the simplicity of exponential functions.

In our specific problem, Abel's Theorem allowed us to find the Wronskian at \(t = 4\) from its known value at \(t = 2\), making it a powerful method for verifying solutions and their properties.

Second-Order Linear Homogeneous Differential Equation

A second-order linear homogeneous differential equation is a type of differential equation characterized by a linear operator applied twice to a function. Consider equations of the form:\[t^2 y'' - 2t y' + (3 + t) y = 0.\]

• The term 'second-order' signifies the highest derivative of the function is two.
• 'Homogeneous' indicates there are no additional terms added to the equation. Usually, the solutions to these equations are functions that best describe a system's behavior in response to no external inputs.

Such equations often appear in physics and engineering, describing systems like mechanical vibrations or electrical circuits. Solving them typically involves finding linearly independent solutions, using techniques like the characteristic equation or employing the Wronskian to verify these solutions.