Small wonder: you leave out ##r## from the derivatives of ##y##. Except in ##C_n##, you want to replace ##n## by ##n+r##yecko said:
You got the recurrence relation for an:yecko said:
The recurrence relation for an isyecko said:
I do not understand what you mean on "summation form"yecko said:
To solve a linear ODE using a power series, you first need to rewrite the ODE in terms of a power series. This involves substituting the power series into the ODE and equating coefficients of like powers. Then, you need to find the recurrence relation by solving for the coefficients. Finally, you can use the initial conditions to determine the values of the coefficients and thus find the solution to the ODE.
A power series is a series of the form ∑n=0∞ an(x-x0)n, where an are the coefficients and x0 is the center of the series. It is an infinite series of terms with increasing powers of x. Power series are often used in calculus to approximate functions.
Solving ODEs using power series allows for the determination of an exact solution, rather than just an approximation. This is useful in situations where an exact solution is necessary, such as in physics and engineering problems. Additionally, power series solutions can be used to find solutions to ODEs that cannot be solved using other methods.
Power series solutions to ODEs are used in a variety of fields, including physics, engineering, and economics. They can be used to model physical systems, such as in the study of fluid dynamics or electrical circuits. Power series solutions are also used in optimization problems, where the goal is to find the maximum or minimum of a function.
One of the main challenges in solving ODEs using power series is finding the recurrence relation for the coefficients. This often involves complex algebraic manipulations and can be difficult for higher-order ODEs. Additionally, power series solutions may not always converge, so it is important to check for convergence when using this method.
