9.3 Cauchy-Euler Equations

Cauchy-Euler equations are a special type of linear differential equations where the variable appears as a power in each term. They pop up in real-world problems like heat conduction in tapered rods and beam vibrations with varying cross-sections.

These equations have a unique structure that allows for specialized solution techniques. By transforming them into linear equations with constant coefficients, we can use the characteristic equation method to solve them, making them a key part of higher-order linear differential equations.

Cauchy-Euler equations

Characteristics and applications

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• Cauchy-Euler equations comprise a special class of linear differential equations with variable coefficients where the variable appears as a power in each term
• Second-order Cauchy-Euler equation general form $ax^2y'' + bxy' + cy = f(x)$ with a, b, and c as constants, and f(x) as a function of x or zero
• Also known as Euler equations or equidimensional equations due to their unique structure
• Arise in problems involving heat conduction in tapered rods, vibrations of beams with varying cross-sections, and certain economic models
• Coefficients of the highest-order derivative term and the independent variable x always have the same power
• Maintain their form under the transformation $x = e^t$, which enables their solution method
• Higher-order Cauchy-Euler equations follow a similar pattern with the general form $a_nx^ny^{(n)} + a_{n-1}x^{n-1}y^{(n-1)} + ... + a_1xy' + a_0y = f(x)$
  • n represents the order of the equation
  • a_n, a_{n-1}, ..., a_1, a_0 are constants
• Unique structure allows for specialized solution techniques
  • Enables transformation into linear equations with constant coefficients
  • Facilitates the use of characteristic equation method

Examples and applications

• Heat conduction in tapered rods
  • Models temperature distribution in non-uniform heat-conducting materials
  • Example equation: $x^2\frac{d^2T}{dx^2} + x\frac{dT}{dx} - 2T = 0$
    • T represents temperature
    • x represents distance along the rod
• Vibrations of beams with varying cross-sections
  • Describes the displacement of a non-uniform beam under stress
  • Example equation: $x^2\frac{d^2y}{dx^2} + 3x\frac{dy}{dx} + y = 0$
    • y represents displacement
    • x represents position along the beam
• Economic models
  • Used in certain growth models and financial calculations
  • Example equation: $x^2\frac{d^2P}{dx^2} + 4x\frac{dP}{dx} - 2P = 0$
    • P represents price or economic variable
    • x represents time or another economic factor

Transformation of Cauchy-Euler equations

Substitution process

• Primary substitution used $x = e^t$ or equivalently $t = \ln(x)$
• Substitution leads to $\frac{dx}{dt} = e^t$ and $\frac{d^2x}{dt^2} = e^t$, crucial in the transformation process
• Apply chain rule to express derivatives with respect to x in terms of derivatives with respect to t
• For second-order Cauchy-Euler equation, transformation yields:
  • $\frac{dy}{dx} = \frac{1}{x}\frac{dy}{dt}$
  • $\frac{d^2y}{dx^2} = \frac{1}{x^2}(\frac{d^2y}{dt^2} - \frac{dy}{dt})$
• After substitution and simplification, resulting equation has constant coefficients in terms of t
• Boundary of original equation (typically x > 0) transforms to $-\infty < t < \infty$ in new equation
• Higher-order Cauchy-Euler equations follow similar transformation pattern
  • Involves more complex applications of the chain rule
  • Results in higher-order linear equations with constant coefficients

Examples of transformation

• Example 1: Transform $x^2y'' + 3xy' - 4y = 0$
  • Substitute $x = e^t$ and apply chain rule
  • Resulting equation: $\frac{d^2y}{dt^2} + 2\frac{dy}{dt} - 4y = 0$
• Example 2: Transform $x^3y''' + 2x^2y'' - xy' + y = 0$
  • Substitute $x = e^t$ and apply chain rule
  • Resulting equation: $\frac{d^3y}{dt^3} + 3\frac{d^2y}{dt^2} + 2\frac{dy}{dt} + y = 0$

Solving Cauchy-Euler equations

Characteristic equation method

• After transformation, equation becomes linear differential equation with constant coefficients
• Form characteristic equation by substituting $y = e^{rt}$ into transformed equation, where r is a constant
• Roots of characteristic equation determine form of general solution in terms of t
• For real and distinct roots, solution is linear combination of $e^{r_it}$, where r_i are the roots
• For repeated real roots, solution includes terms of form $t^k e^{rt}$, where k ranges from 0 to (multiplicity - 1)
• For complex conjugate roots a ± bi, solution includes terms of form $e^{at}(c_1\cos(bt) + c_2\sin(bt))$
• Find particular solution for non-homogeneous equations using methods (undetermined coefficients, variation of parameters)
• Back-substitute $t = \ln(x)$ to express solution in terms of original variable x

Solution examples

• Example 1: Solve $x^2y'' + 3xy' - 4y = 0$
  • Transformed equation: $\frac{d^2y}{dt^2} + 2\frac{dy}{dt} - 4y = 0$
  • Characteristic equation: $r^2 + 2r - 4 = 0$
  • Roots: $r = -3$ or $r = 1$
  • General solution: $y = c_1e^{-3t} + c_2e^t$
  • Back-substitute: $y = c_1x^{-3} + c_2x$
• Example 2: Solve $x^2y'' + 5xy' + 4y = 0$
  • Transformed equation: $\frac{d^2y}{dt^2} + 4\frac{dy}{dt} + 4y = 0$
  • Characteristic equation: $r^2 + 4r + 4 = 0$
  • Repeated root: $r = -2$
  • General solution: $y = (c_1 + c_2t)e^{-2t}$
  • Back-substitute: $y = (c_1 + c_2\ln(x))x^{-2}$

General solutions vs initial conditions

General solution properties

• General solution expressed as linear combination of fundamental solutions obtained from characteristic equation method
• For second-order equation, general solution takes form $y = c_1y_1(x) + c_2y_2(x)$, where y_1(x) and y_2(x) are linearly independent solutions
• Fundamental solutions for Cauchy-Euler equations often involve terms:
  • $x^r$
  • $x^r\ln(x)$
  • $x^a(c_1\cos(b\ln(x)) + c_2\sin(b\ln(x)))$
• Higher-order Cauchy-Euler equations follow similar pattern with more terms in general solution

Applying initial conditions

• Find particular solution by applying initial conditions to general solution and its derivatives at specified point (typically x = 1 for convenience)
• Form system of linear equations using initial conditions to solve for constants c_1, c_2, etc.
• Uniqueness theorem for linear differential equations ensures unique solution exists for well-posed initial value problems
• Higher-order Cauchy-Euler equations involve more constants and initial conditions
• Verify final solution by substituting back into original Cauchy-Euler equation
  • Check satisfaction of both equation and initial conditions

Examples with initial conditions

• Example 1: Solve $x^2y'' + 3xy' - 4y = 0$ with y(1) = 2 and y'(1) = 3
  • General solution: $y = c_1x^{-3} + c_2x$
  • Apply initial conditions:
    • $2 = c_1 + c_2$
    • $3 = -3c_1 + c_2$
  • Solve system of equations:
    • $c_1 = -1$, $c_2 = 3$
  • Particular solution: $y = -x^{-3} + 3x$
• Example 2: Solve $x^2y'' + 5xy' + 4y = 0$ with y(1) = 1 and y'(1) = 0
  • General solution: $y = (c_1 + c_2\ln(x))x^{-2}$
  • Apply initial conditions:
    • $1 = c_1$
    • $0 = c_2 - 2c_1$
  • Solve system of equations:
    • $c_1 = 1$, $c_2 = 2$
  • Particular solution: $y = (1 + 2\ln(x))x^{-2}$