- #1
Andreol263
- 77
- 15
Why these ODEs when applied some boundary conditions, like x = 0, their solution of the form Ax^k + Bx^(-k), B WILL have to go to zero?Like some problems which involve spherical harmonics...
An ODE (ordinary differential equation) of the form axny(n) + bx(n-1)y(n-1) + ... + cx'y + dy = 0, where a, b, c, and d are constants and n is a non-negative integer.
Question 2: How do you solve a Cauchy-Euler ODE?To solve a Cauchy-Euler ODE, you can use the substitution y = xm. This will transform the ODE into a polynomial equation, which can then be solved using methods such as factoring or the quadratic formula.
Question 3: What is the characteristic equation in a Cauchy-Euler ODE?The characteristic equation in a Cauchy-Euler ODE is the polynomial equation that results from substituting y = xm into the ODE. It is used to find the values of m that will yield a solution to the ODE.
Question 4: Can a Cauchy-Euler ODE have non-integer exponents?Yes, a Cauchy-Euler ODE can have non-integer exponents. In this case, the substitution y = xm is still used, but the characteristic equation will result in complex roots, which will yield complex solutions for y.
Question 5: What are some real-world applications of Cauchy-Euler ODEs?Cauchy-Euler ODEs can be used to model various physical phenomena, such as the motion of a pendulum, the decay of radioactive materials, and the flow of fluids. They are also commonly used in engineering and economics to model systems with changing variables over time.