Question

Let \(y_{1}(t)=t^{\gamma} \cos (\delta \ln t)\) and \(y_{2}(t)=t^{\gamma} \sin (\delta \ln t), t>0\), where \(\delta\) and \(\gamma\) are real constants with \(\delta \neq 0\). Solutions of this form arise when the characteristic equation has complex roots. Compute the Wronskian of this pair of solutions, and show that they form a fundamental set of solutions on \(0

Short answer

In summary, we calculated the Wronskian of the two given solutions \(y_1(t) = t^{\gamma} \cos(\delta \ln t)\) and \(y_2(t) = t^{\gamma} \sin(\delta \ln t)\) and showed that it is non-zero for all \(t > 0\). Thus, \(y_1(t)\) and \(y_2(t)\) form a fundamental set of solutions on the interval \((0, \infty)\).

Step-by-step solution

Calculate the derivatives of \(y_1(t)\) and \(y_2(t)\)

First, we need to calculate the derivatives of \(y_1(t)\) and \(y_2(t)\): \(y_1(t) = t^{\gamma} \cos(\delta \ln t)\) \(y_1'(t) = \frac{d}{dt}(t^{\gamma} \cos(\delta \ln t))\) Similarly for \(y_2(t)\): \(y_2(t) = t^{\gamma} \sin(\delta \ln t)\) \(y_2'(t) = \frac{d}{dt}(t^{\gamma} \sin(\delta \ln t))\) Now, let's compute the derivatives using the product rule and chain rule: \(y_1'(t) = \gamma t^{\gamma-1} \cos(\delta \ln t) - \delta t^{\gamma - 1} \sin(\delta \ln t)\) \(y_2'(t) = \gamma t^{\gamma-1} \sin(\delta \ln t) + \delta t^{\gamma - 1} \cos(\delta \ln t)\)

Calculate the Wronskian of \(y_1(t)\) and \(y_2(t)\)

Now, let's compute the Wronskian using the formula \(W(y_1, y_2) = |y_1'(t) * y_2(t) - y_1(t) * y_2'(t)|\): \(W(y_1, y_2) = |\gamma t^{2(\gamma-1)} (\cos^2(\delta \ln t) + \sin^2(\delta \ln t)) - \delta^2 t^{2(\gamma-1)} (\sin^2(\delta \ln t) * \cos^2(\delta \ln t))|\) Rewriting the terms: \(W(y_1, y_2) = t^{2(\gamma-1)}(\gamma^2 - \delta^2\sin^2(\delta \ln t) * \cos^2(\delta \ln t))\)

Show that the Wronskian is non-zero for \(0

Now, we need to prove that the Wronskian is non-zero for the given interval \((0, \infty)\). Notice that \(t^{2(\gamma-1)} > 0\) for all \(t > 0\). Also, \((\gamma^2 - \delta^2\sin^2(\delta \ln t) * \cos^2(\delta \ln t))\) is greater than or equal to 0, since \(\gamma^2 > \delta^2 * \sin^2(\delta \ln t) * \cos^2(\delta \ln t)\) (recall that \(\delta \neq 0\) and both \(\gamma\) and \(\delta\) are real constants). Since the Wronskian is non-zero for all \(t > 0\), the given solutions \(y_1(t)\) and \(y_2(t)\) form a fundamental set of solutions on the interval \((0, \infty)\).

Key concepts

Fundamental Set of Solutions

A fundamental set of solutions is a collection of solutions to a differential equation that can be used to express any solution to that equation as a linear combination. Consider a linear differential equation of the form \(y''(t) + p(t)y'(t) + q(t)y(t) = 0\). Finding a fundamental set involves identifying two solutions that are linearly independent.

For solutions to be linearly independent, their Wronskian must be non-zero. This means that at no point should the solutions be able to be expressed as a scalar multiple of one another. In the given exercise, the functions \(y_1(t)\) and \(y_2(t)\) are examined using their derivatives and the Wronskian is computed. The Wronskian expression turns out to be \(t^{2(\gamma-1)}\), which is non-zero for all \(t > 0\) assuming \(\gamma^2 - \delta^2\sin^2(\delta \ln t) * \cos^2(\delta \ln t)\ge 0\).

Therefore, \(y_1(t)\) and \(y_2(t)\) form a fundamental set of solutions on \((0, \infty)\), confirming that any solution to the differential equation in question can be expressed as a linear combination of \(y_1(t)\) and \(y_2(t)\).

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. These equations are fundamental in describing various physical phenomena, from heat transfer to wave propagation.

The specific type of differential equation in this exercise is a second-order linear homogeneous equation. This means there's an order of two, relating a function to its second derivative without any separate terms (homogeneous). The general form is \(a y'' + b y' + cy = 0\), where \(a, b,\) and \(c\) are constants or functions.

The solutions to these equations can often be found using characteristic equations, which transform the differential equation into a polynomial. The roots of this polynomial help determine the type of solutions. In linear equations with constant coefficients, we might discover real distinct, repeated roots, or complex roots, each leading to different types of general solutions.

Complex Roots

Complex roots emerge from the characteristic equation of a differential equation when solving for \(r\) in \(ar^2 + br + c = 0\), resulting in non-real numbers. These complex roots indicate oscillatory behavior in the solutions.

For complex roots in the form \(\alpha \pm \beta i\), the general solution involves exponential and trigonometric functions: \(y(t) = e^{\alpha t}(C_1 \cos(\beta t) + C_2 \sin(\beta t))\). This reflects how the function oscillates and decays or grows according to \(\alpha\) and \(\beta\).

In our context, the solutions \(y_1(t) = t^{\gamma}\cos(\delta \ln t)\) and \(y_2(t) = t^{\gamma} \sin(\delta \ln t)\) arise when the roots of the characteristic equation are complex. The presence of \(\cos(\delta \ln t)\) and \(\sin(\delta \ln t)\) components illustrates the typical oscillating nature seen with complex roots, adapting through a logarithmic time scale influenced by \(\delta\) and \(\gamma\).