Problem 12 - Advanced Engineering Mathematics

Question

FindaCauchy-Eulerdifferentialequation \(a x^{2} y^{\prime \prime}+b x y^{\prime}+c y=0\), where \(a, b\), and \(c\) are real constants, if it is known that (a) \(m_{1}=3\) and \(m_{2}=-1\) are roots of its auxiliary equation, (b) \(m_{1}=i\) is a complex root of its auxiliary equation.

Step-by-Step Solution

Answer

(a) The equation is \(x^2 y'' - 2xy' - 3y = 0\). (b) The equation is \(x^2 y'' + y = 0\).

Step 1: Understand the Cauchy-Euler Differential Equation

The Cauchy-Euler differential equation has the form \( ax^2 y'' + bxy' + cy = 0 \). It's characterized by its form, where the coefficients of the derivatives are powers of \( x \). The equation's order is determined by the highest derivative present, in this case, the second derivative \( y'' \).

Step 2: Form the Auxiliary Equation

For Cauchy-Euler differential equations, the auxiliary equation is obtained by assuming a solution of the form \( y = x^m \). Substituting this into the differential equation gives \( am(m-1)x^m + bmx^m + cx^m = 0 \), leading to the auxiliary equation \( am^2 + (b-a)m + c = 0 \).

Step 3: Solve for Specific Roots (Case A)

Given roots \( m_1 = 3 \) and \( m_2 = -1 \), the auxiliary equation becomes \((m - 3)(m + 1) = 0\), which expands to \(m^2 - 3m + 1m - 3 = m^2 - 2m - 3 = 0\). Multiplying out gives the equation \( am^2 + (b-a)m + c = a(m^2 - 2m - 3) = 0 \). Thus, identifying coefficients with the general form yields \( a = 1, b = -2, c = -3 \).

Step 4: Solve for Complex Root (Case B)

For a single complex root \( m_1 = i \), its conjugate \( m_2 = -i \) must also be a root, making the quadratic equation \((m - i)(m + i) = m^2 + 1 = 0\). Therefore, the auxiliary equation \( am^2 + (b-a)m + c = 0 \) becomes \(am^2 + (b-a)m + c = a(m^2 + 1) = 0 \), where \( a = 1 \), \( b-a = 0 \), and \( c = 1 \). Solving these yield \( b = 0 \) and \( c = a = 1 \).

Step 5: Verify and Compile the Solutions

For (a), we found the differential equation as \( x^2 y'' - 2xy' - 3y = 0 \). For (b), the equation is \( x^2 y'' + y = 0 \). We ensure these satisfy the conditions of having the specified roots.

Key Concepts

Auxiliary Equation

In the context of Cauchy-Euler differential equations, the auxiliary equation plays a crucial role in understanding the behavior and solutions of the differential equation. The Cauchy-Euler differential equation is given by:\[ax^2 y'' + bxy' + cy = 0\]To derive the auxiliary equation, we begin by assuming a solution of the form \(y = x^m\). Substituting \(y = x^m\) and its derivatives into the original differential equation leads us to:\[am(m-1)x^m + bmx^m + cx^m = 0\]After factoring out \(x^m\), since \(x^m eq 0\), the terms simplify into a quadratic equation:\[am^2 + (b-a)m + c = 0\]This quadratic equation is known as the auxiliary equation. The task becomes solving this quadratic to find the roots, which help in formulating the general solution of the differential equation.

Roots of Differential Equations

The roots of the auxiliary equation are essential in determining the solutions of the associated differential equation. The nature of these roots—in particular, their real and complex characteristics—dictates the form of the solutions.

• Real Roots: If the auxiliary equation's roots are real and distinct, like in part (a) where \(m_1 = 3\) and \(m_2 = -1\), the general solution is expressed as a linear combination of power functions:\[y(x) = c_1 x^{m_1} + c_2 x^{m_2}\]
• Repeated Real Roots: If the roots are real and repeated, then an extra factor of \(\ln(x)\) is introduced in the solution to account for this repetition.
• Complex Roots: For complex roots, such as in part (b) where \(m_1 = i\) and its conjugate \(m_2 = -i\) appear, the solution involves trigonometric functions. This is due to the Euler's formula which relates complex exponentials and trigonometric functions.

Understanding these roots thoroughly is key, as they directly form the blueprint of how the solution behaves over the domain of the differential equation.

Complex Roots

Complex roots arise when the auxiliary equation, derived from the Cauchy-Euler differential equation, has non-real solutions. For example, when given a root \(m_1 = i\), its complex conjugate \(m_2 = -i\) must also be a root of the equation.Complex roots yield solutions that are inherently connected to trigonometric and exponential functions due to Euler’s formula. Specifically, if you have complex roots \(m = \, \pm i\), the general solution can be articulated as:\[y(x) = x^{\text{Re}(m)} (c_1 \cos(\ln(x)) + c_2 \sin(\ln(x)))\]Given that \(\text{Re}(i) = 0\), the expression simplifies further in linear form. The connection with trigonometric functions like cosine and sine emerges due to the imaginary part of the root. As seen in the problem's part (b), this leads to significant changes to the structure of the solutions, highlighting the versatile nature of differential equation solutions when impacted by complex roots.

Second-order Differential Equations

Second-order differential equations are a fundamental class of differential equations characterized by the presence of the highest derivative being of second order. The Cauchy-Euler form \( ax^2 y'' + bxy' + cy = 0 \) is a classic example. Here, \(y''\) is the second derivative of \(y\) with respect to \(x\).Key features of second-order differential equations include their capability to model a wide range of physical phenomena, from mechanical vibrations and electrical circuits to heat transfer problems. The solutions to these equations provide insight into the dynamics of the systems they model.In Cauchy-Euler equations, the order of the equation informs the auxiliary equation formation, which then guides the solution process. Typically, the auxiliary equation, being quadratic, results in two roots. Such roots dictate if solutions are exponential, polynomial, or trigonometric in nature. Understanding the process of finding and interpreting these roots is crucial in the broader scope of analyzing and solving second-order differential equations effectively.