Problem 37 - Advanced Engineering Mathematics

Question

Find a homogeneous Cauchy-Euler differential equation whose general solution is given. $$ y=c_{1} \cos (\ln x)+c_{2} \sin (\ln x) $$

Step-by-Step Solution

Answer

The homogeneous Cauchy-Euler equation is \(x^2 y'' + y = 0\).

Step 1: Understanding the Given Solution

The given general solution is \(y = c_1 \cos (\ln x) + c_2 \sin (\ln x)\). This indicates that the solution involves logarithmic trigonometric functions, suggesting a Cauchy-Euler equation or an equation with variable coefficients.

Step 2: Assuming the Form of the Differential Equation

Cauchy-Euler equations are typically of the form \(a x^2 y'' + b x y' + c y = 0\). Since the solution involves trigonometric functions of \(\ln x\), the characteristic equation corresponds to \( r^2 + 1 = 0 \) for a simplified version of this equation.

Step 3: Solving the Characteristic Equation

The characteristic equation derived is \( r^2 + 1 = 0 \), which occurs by identifying that solutions of the form \(\cos(\ln x)\) and \(\sin(\ln x)\) arise from roots \( r = i \) and \( r = -i \), indicative of a second order differential equation.

Step 4: Formulating the Cauchy-Euler Differential Equation

Using the standard form for a Cauchy-Euler equation with the calculated characteristic roots, we derive \( x^2 y'' + y = 0 \). This differential equation has solutions matching the given form \(y = c_1 \cos (\ln x) + c_2 \sin (\ln x)\).

Key Concepts

Homogeneous Differential Equation

A homogeneous differential equation is a type of differential equation in which every term is a function of the dependent variable and its derivatives. Typically, these equations equal zero. They can be expressed in the form:

• For first order: \( f(x, y, y') = 0 \)
• For second order: \( a(x) y'' + b(x) y' + c(x) y = 0 \)

Key features of homogeneous differential equations include:

• Linearly combining solutions due to their nature being zero on one side, allowing solutions like \( y = c_1 y_1 + c_2 y_2 \).
• The solution can often involve constant coefficients like \( c_1 \) and \( c_2 \) that reflect arbitrary constants determined by initial or boundary conditions.

Understanding these equations requires practice in identifying and solving characteristic equations, which reveal specific solution forms depending on the roots obtained.

Logarithmic Trigonometric Functions

Logarithmic trigonometric functions combine logarithms and trigonometric expressions, often appearing in solutions to differential equations involving variable coefficients. An example from the given problem is \( \cos(\ln x) \) and \( \sin(\ln x) \).
These functions come into play particularly when dealing with Cauchy-Euler equations or similar forms where the independent variable transformations, such as substitution with \( z = \ln x \), provide a bridge to simpler trigonometric solutions. Here’s how they work:

• Substitute \( x \) with an exponential function involving \( z = \ln x \), making calculations simpler.
• Use identities such as \( \cos(a + b) = \cos a \cos b - \sin a \sin b \) to solve complex expressions or differentiate them.
• Common in physics and engineering, serving to model oscillations where frequency logarithmically varies.

Characteristic Equation

The characteristic equation is a polynomial equation that arises in the process of finding solutions to linear differential equations, especially those with constant coefficients. It transforms the original differential equation into an algebraic one, making it easier to solve for the roots, which indicate the form of the solutions.
For a Cauchy-Euler differential equation, the characteristic equation is determined by:

• Assuming solutions in the form \( y = x^r \), leading to substituting these into the equation.
• The example characteristic equation derived: \( r^2 + 1 = 0 \) yields roots \( r = i \) and \( r = -i \), complex roots reflecting sinusoidal functions.
• These roots directly relate to the general solution involving sine and cosine functions of the logarithm of \( x \).

The ability to derive and solve characteristic equations is crucial for understanding the behavior of linear differential systems.

Variable Coefficients

Variable coefficients in differential equations mean the coefficients of the dependent variable and its derivatives are not constant but vary with the independent variable. An equation with variable coefficients looks like:

• \( a(x) y'' + b(x) y' + c(x) y = 0 \)

Cauchy-Euler equations are a classic example where coefficients are powers of the independent variable, usually expressed as:

• \( ax^2 y'' + bxy' + cy = 0 \)

The challenges with variable coefficients include:

• Transforming such equations through substitutions can convert them to constant coefficient forms, making them easier to solve.
• These require sophisticated knowledge of differential equation theory and often specific methods like power series or variation of parameters to solve.
• Technological applications range from mechanical systems to oscillatory circuits where system properties change over time or position.

Recognizing that an equation has variable coefficients is crucial in determining the proper method for solving it.