General solution to the euler-cauchy equation

In summary, the Euler-Cauchy equation is a second-order linear differential equation with the form x^2y'' + pxy' + qy = 0. The general solution depends on the values of p and q, with different forms for real and distinct values, complex conjugates, and initial conditions. It can be applied to real-world problems and has two special cases when p or q equals 0, resulting in simpler forms of the general solution.
  • #1
asdf1
734
0
why does the general solution to the euler-cauchy equation only work for x>0?
 
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  • #2
It doesn't. It may well work for x< 0. Certainly, because the general equation has a singularity at x= 0, we can't expect a general solution to exist at x= 0 or be extended past x= 0- but you can have solutions that are valid for x> 0 and solutions that are valid for x< 0.
 
  • #3
ok, that makes sense~ :)
 

1. What is the Euler-Cauchy equation?

The Euler-Cauchy equation is a type of second-order linear differential equation that can be written in the form x^2y'' + pxy' + qy = 0, where p and q are constants.

2. What is the general solution to the Euler-Cauchy equation?

The general solution to the Euler-Cauchy equation depends on the values of p and q. If p and q are real and distinct, the general solution is y(x) = c1x^r1 + c2x^r2, where r1 and r2 are the roots of the auxiliary equation r^2 + (p-1)r + q = 0. If p and q are complex conjugates, the general solution is y(x) = c1x^r1cos(ax) + c2x^r2sin(ax), where r1 and r2 are the real parts of the roots of the auxiliary equation and a is the imaginary part divided by 2.

3. How do I solve the Euler-Cauchy equation with initial conditions?

To solve the Euler-Cauchy equation with initial conditions, you can first find the general solution and then use the initial conditions to determine the values of the constants c1 and c2. Alternatively, you can use the method of undetermined coefficients to find a particular solution that satisfies the initial conditions.

4. Can the Euler-Cauchy equation be applied to real-world problems?

Yes, the Euler-Cauchy equation can be applied to real-world problems in physics, engineering, and other fields. It is commonly used in mechanics and electrical circuits, among other applications.

5. Are there any special cases of the Euler-Cauchy equation?

Yes, there are two special cases of the Euler-Cauchy equation. The first is when p = 0, which reduces the equation to the form x^2y'' + qy = 0. The second is when q = 0, which reduces the equation to the form x^2y'' + pxy' = 0. These cases have simpler forms of the general solution compared to the general case.

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