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why does the general solution to the euler-cauchy equation only work for x>0?
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.
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.
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.
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.
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.