The Euler-Cauchy equation, also known as the equidimensional or Euler's differential equation, is a fundamental type of second-order linear differential equation that appears frequently in mathematical modeling, physics, and engineering. Its unique structure allows for solutions that involve power functions, making it a vital topic for students and professionals dealing with differential equations. Understanding how to solve the Euler-Cauchy equation equips you with a powerful tool for tackling various scientific problems efficiently.
How to Solve Euler Cauchy Equation
The general form of the Euler-Cauchy equation is:
ax² y'' + bx y' + c y = 0
where a, b, and c are constants, and y is a function of x.
Solving this differential equation involves transforming it into a constant-coefficient differential equation, which is easier to solve. The standard method involves assuming a solution of the form y = x^m, where m is a constant to be determined.
Step-by-Step Method to Solve the Euler Cauchy Equation
1. Assume a Power-Law Solution
Start by assuming a solution of the form:
- y = x^m
where m is an unknown constant. This form is chosen because the coefficients involve powers of x, and the derivatives of y = x^m are straightforward to compute:
- y' = m x^{m-1}
- y'' = m (m - 1) x^{m-2}
2. Substitute into the Differential Equation
Substitute y, y', and y'' into the original equation:
ax² [m (m - 1) x^{m-2}] + bx [m x^{m-1}] + c x^m = 0
Simplify each term:
a m (m - 1) x^{m} + b m x^{m} + c x^{m} = 0
Factoring out x^m (which is non-zero for x ≠ 0), we get:
x^{m} [a m (m - 1) + b m + c] = 0
This leads to the characteristic (or indicial) equation:
a m (m - 1) + b m + c = 0
3. Solve the Characteristic Equation
The quadratic equation in m:
a m^2 + (b - a) m + c = 0
can be solved using the quadratic formula:
m = [-(b - a) ± √((b - a)^2 - 4 a c)] / (2 a)
Depending on the discriminant, the roots can be real and distinct, real and repeated, or complex conjugates:
- Discriminant > 0: Two distinct real roots.
- Discriminant = 0: Repeated real root.
- Discriminant < 0: Complex conjugate roots.
4. Write the General Solution
Based on the roots of the characteristic equation:
- Two distinct real roots m₁ and m₂:
The general solution is:
y(x) = C₁ x^{m₁} + C₂ x^{m₂}
The general solution is:
y(x) = C₁ x^{m} + C₂ x^{m} \ln x
The general solution becomes:
y(x) = x^{α} [A \cos(β \ln x) + B \sin(β \ln x)]
where C₁, C₂, A, and B are arbitrary constants determined by initial conditions.
Examples of Solving Euler Cauchy Equation
Example 1: Solve x^{2} y'' + 3 x y' + 2 y = 0
Identify coefficients:
- a = 1, b = 3, c = 2
Set up the characteristic equation:
m^2 + (3 - 1) m + 2 = 0
m^2 + 2 m + 2 = 0
Calculate discriminant:
Δ = 2^2 - 4 * 1 * 2 = 4 - 8 = -4
Since Δ < 0, roots are complex:
m = [-2 ± √(-4)] / 2 = [-2 ± 2 i] / 2 = -1 ± i
The general solution is:
y(x) = x^{-1} [A \cos(\ln x) + B \sin(\ln x)]
Example 2: Solve x^{2} y'' - 4 x y' + 6 y = 0
Identify coefficients:
- a = 1, b = -4, c = 6
Characteristic equation:
m^2 + (-4 - 1) m + 6 = 0
m^2 - 5 m + 6 = 0
Solve quadratic:
m = [5 ± √(25 - 24)] / 2 = [5 ± 1] / 2
Roots:
- m₁ = (5 + 1)/2 = 3
- m₂ = (5 - 1)/2 = 2
Solution:
y(x) = C₁ x^{3} + C₂ x^{2}
Key Points to Remember When Solving Euler Cauchy Equations
- The solution involves assuming y = x^m, transforming the differential equation into a quadratic characteristic equation.
- The roots of the characteristic equation determine the form of the general solution.
- Complex roots produce oscillatory solutions involving sine and cosine functions with a logarithmic argument.
- Repeated roots lead to solutions with an additional logarithmic term, i.e., y = x^{m} and y = x^{m} \ln x.
- Always check the discriminant to determine the nature of the roots and the corresponding form of the solution.
By following these steps and understanding the underlying principles, you can confidently solve Euler-Cauchy equations encountered in various mathematical and engineering contexts. Mastery of this method simplifies solving many problems that initially seem complex, providing a strong foundation for further study of differential equations.