Problem 14 - Advanced Engineering Mathematics

Question

Solve the given differential equation. $$ x^{2} y^{\prime \prime}-7 x y^{\prime}+41 y=0 $$

Step-by-Step Solution

Answer

The solution is \( y = x^4(C_1 \cos(5 \ln x) + C_2 \sin(5 \ln x)) \).

Step 1: Identify the Type of Differential Equation

The given differential equation is a second-order linear homogeneous differential equation with variable coefficients. It can be classified as a Cauchy-Euler equation, also known as an equidimensional equation.

Step 2: Assume a Solution Form

For Cauchy-Euler equations, we assume a solution of the form \( y = x^m \). This assumption is made because these types of equations can often be converted into a simpler form using substitution.

Step 3: Compute Derivatives

Calculate the first and second derivatives of \( y = x^m \): \( y' = mx^{m-1} \) and \( y'' = m(m-1)x^{m-2} \). These will be substituted into the differential equation next.

Step 4: Substitute Derivatives into the Equation

Replace \( y \), \( y' \), and \( y'' \) in the original equation with the assumed forms: \( x^2(m(m-1)x^{m-2}) - 7x(mx^{m-1}) + 41x^m = 0 \). Simplify this to get: \( x^m[m(m-1) - 7m + 41] = 0 \).

Step 5: Formulate the Characteristic Equation

The characteristic equation is obtained from the simplified expression: \( m(m-1) - 7m + 41 = 0 \), which simplifies to \( m^2 - 8m + 41 = 0 \).

Step 6: Solve the Characteristic Equation

Use the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1, b = -8, c = 41 \). The discriminant \( b^2 - 4ac = (-8)^2 - 4 \times 1 \times 41 = 64 - 164 = -100 \), indicating complex roots.

Step 7: Calculate Complex Roots

The roots are \( m = \frac{8 \pm i\sqrt{100}}{2} = 4 \pm 5i \). These are complex conjugates: \( m_1 = 4 + 5i \) and \( m_2 = 4 - 5i \).

Step 8: Write General Solution

The general solution of the differential equation with complex roots \( \alpha \pm \beta i \) is \( y = x^{\alpha}(C_1 \cos(\beta \ln x) + C_2 \sin(\beta \ln x)) \). Thus, \( y = x^4(C_1 \cos(5 \ln x) + C_2 \sin(5 \ln x)) \).

Key Concepts

Second-order Differential Equation

A second-order differential equation involves the second derivative of a function. It is written in the general form: \[ a(x)y'' + b(x)y' + c(x)y = 0 \] where \( y'' \) is the second derivative of \( y \), \( y' \) is the first derivative, and \( a(x), b(x), c(x) \) are coefficients that might be functions of \( x \).
These types of equations model a wide range of phenomena, such as mechanical vibrations, electrical circuits, and even population dynamics. For second-order equations, the solution often involves finding a function \( y(x) \) that satisfies the equation for all \( x \).
In the case of a Cauchy-Euler equation, we have a specific type of second-order equation where the coefficients are proportional to powers of \( x \). The provided problem is such an example: \[ x^2y'' - 7xy' + 41y = 0 \] Here, since all the coefficients are of the form \( x^n \), the equation is identified as a Cauchy-Euler equation, allowing us to find solutions where the function \( y \) takes on a form based on powers of \( x \).

Complex Roots

In solving second-order differential equations, particularly Cauchy-Euler types, we often encounter characteristic equations that have complex roots.
Complex roots arise when the discriminant \( b^2 - 4ac \) of the characteristic equation is negative. This results in roots that involve imaginary numbers.
For our equation, this step results in finding roots that are complex conjugates: \[ m = 4 \pm 5i \]
When dealing with complex roots \( \alpha \pm \beta i \) , the general solution to the differential equation is expressed as: \[ y = x^\alpha (C_1 \cos(\beta \ln x) + C_2 \sin(\beta \ln x)) \] This solution showcases the behavior of the differential equation, encapsulating both exponential growth (from \( x^{\alpha} \)) and oscillatory components (from the sine and cosine terms). This combination reveals the dynamic interplay of trends represented by the complex roots.

Characteristic Equation

The characteristic equation is a crucial concept when solving differential equations, especially in the context of Cauchy-Euler equations.
For these types of equations, the characteristic equation is derived from substituting an assumed solution into the original differential equation. In the case of our problem, assuming a solution of the form \( y = x^m \) allows us to transform the problem into one of finding the roots of a polynomial expression.
In our specific exercise, the characteristic equation emerges as: \[ m^2 - 8m + 41 = 0 \] Solving this quadratic equation involves using the quadratic formula: \[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, the coefficients are \( a = 1, b = -8, \) and \( c = 41 \), with a negative discriminant indicating complex solutions.
Characteristic equations enable us to determine the general form of the solution for Cauchy-Euler differential equations. This critical step allows us to interpret the behavior of the differential system and how it evolves over the domain of interest.