Help with Solving a Cauchy-Euler Differential Equation

In summary, the conversation discusses solving a Cauchy-Euler equation with the substitution of y=ln(x) and using the definition of exponentiation to remove imaginary numbers from the solution.
  • #1
Jim4592
49
0

Homework Statement



x2 y'' + x y' + 4 y = 0


Homework Equations



y = xr
y' = r xr-1
y'' = (r2-r)xr-2

The Attempt at a Solution



x2{(r2-r)xr-1} + x{r xr-1} + 4xr

r2 - r + r + 4

r2 + 4 = 0

r = +- 2i

y = C1x2i + c2x-2i

My question is how can i remove the imaginary number with cos[] and sin[]
If you could be as descriptive as possible i'd really appreciate it!

Thanks in advanced!
 
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  • #2
Use the definition of exponentiation: xa=ea log x.
 
  • #3
In fact, the substitution y= ln(x) will change any "Cauchy-Euler" equation into an equation with constant coefficients with the same characteristic equation. An equation with constant coefficients, with [itex]\pm 2[/itex] as characteristic roots, has general solution C cos(2y)+ D sin(2y).
 

1. What is a Cauchy-Euler differential equation?

A Cauchy-Euler differential equation is a type of second-order ordinary differential equation that can be written in the form ax^2y'' + bxy' + cy = 0, where a, b, and c are constants and y is the unknown function. It is also known as an equidimensional or homogeneous differential equation.

2. How do you solve a Cauchy-Euler differential equation?

To solve a Cauchy-Euler differential equation, you can use the method of substitution. This involves substituting y = x^m into the equation, where m is a constant to be determined. This will reduce the equation to a polynomial in x, which can then be solved using standard methods.

3. What are the conditions for a Cauchy-Euler differential equation to have a solution?

A Cauchy-Euler differential equation must have constant coefficients and must be of the form ax^2y'' + bxy' + cy = 0 in order to have a solution. Additionally, the equation must have non-zero solutions for both y and y'.

4. Can a Cauchy-Euler differential equation have complex solutions?

Yes, a Cauchy-Euler differential equation can have complex solutions. This can happen when the roots of the characteristic equation (the equation obtained by substituting y = x^m) are complex numbers.

5. Are there any real-world applications of Cauchy-Euler differential equations?

Yes, Cauchy-Euler differential equations have many real-world applications in physics, engineering, and other fields. For example, they can be used to model the behavior of a mass-spring system or a vibrating string. They are also used in circuit analysis, heat transfer, and fluid mechanics.

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