# Find the series solution,power series

• dp182
In summary, the given content discusses solving a differential equation involving indicial equations and recurrence relations. The main focus is on finding the roots of the indicial equation and using them to determine the recurrence relation. The conversation also includes a step-by-step explanation of the process and some helpful tips for solving the problem.
dp182

## Homework Statement

2xy''-x(x-1)y'-y=0 about x=0
what are the roots of the indicial equation and for the roots find the recurrence relation that defines the the coef an

## Homework Equations

2xy''-x(x-1)y'-y=0 about x=0

assuming the solution has the form y=$\Sigma$anxn+r
y'=$\Sigma$(n+r)anxn+r-1
y''=$\Sigma$(n+r)(n+r-1)anxn+r-2

## The Attempt at a Solution

after plugging into the solution I get
2$\Sigma$(n+r)(n+r-1)anxn+r-1-$\Sigma$(n+r)anxn+r+1-$\Sigma$(n+r)anxn+r-1-$\Sigma$anxn+r
then I attempt to make all the x's the same and and make the sigma's equal so after doing that I get
2$\Sigma$(n+r+1)(n+r)an+1xn+r-$\Sigma$(n+r-1)an-1xn+r-$\Sigma$(n+r+1)an+1xn+r-$\Sigma$anxn+r
I know that I need to replace the 0 under the sigma's with a (-1) on terms 1,3 but term 2 is what's throwing me off any help would be great

The third sum should be positive. Explicitly writing in the limits, you have
$$2\sum_{n=0}^\infty (n+r)(n+r-1)a_n x^{n+r-1} -\sum_{n=0}^\infty (n+r) a_n x^{n+r+1} +\sum_{n=0}^\infty (n+r)a_n x^{n+r-1} -\sum_{n=0}^\infty a_n x^{n+r} = 0$$
As you noted, only the first and third sums give you a $x^{r-1}$ term, and the second sum doesn't give you an $x^r$ term. Separating those terms out, you have
\begin{eqnarray*}
&&[2r(r-1) + r]a_0x^{r-1} + \\
&&[2(r+1)r a_1 + (r+1)a_1 - a_0]x^r + \\
&&2\sum_{n=1}^\infty (n+r+1)(n+r)a_{n+1}x^{n+r}
-\sum_{n=1}^\infty (n+r-1)a_{n-1}x^{n+r}
+\sum_{n=1}^\infty (n+r+1)a_{n+1}x^{n+r}
-\sum_{n=1}^\infty a_n x^{n+r} = 0
\end{eqnarray*}
By assumption, a0 isn't equal to 0, so you must have 2r(r-1)+r=0. That's your indicial equation.

## 1. What is a series solution and why is it important?

A series solution is a mathematical technique used to approximate the solution to a differential equation. It involves expressing the solution as a sum of terms in a particular sequence, called a power series. It is important because it allows us to find approximate solutions to equations that cannot be solved analytically.

## 2. How do you find the series solution for a given equation?

To find the series solution for a given equation, the first step is to write the equation in the form of a power series. Next, we determine the coefficients of the power series by using the initial conditions or boundary conditions of the equation. Finally, we combine the coefficients with the power series to obtain the series solution.

## 3. What is the difference between a power series and a Taylor series?

A power series is a type of infinite series where each term is a polynomial function of the variable x. A Taylor series is a specific type of power series that is centered at a particular point and has the coefficients determined by the derivatives of the function at that point. While all Taylor series are power series, not all power series are Taylor series.

## 4. Can series solutions be used to solve any type of differential equation?

No, series solutions can only be used to solve linear ordinary differential equations. Nonlinear equations or partial differential equations require different methods for finding solutions.

## 5. What are some applications of series solutions in real-world problems?

Series solutions have many practical applications in fields such as physics, engineering, and finance. They can be used to model physical phenomena, approximate solutions to differential equations in engineering problems, and evaluate financial investments. Additionally, series solutions are used in computer algorithms for solving differential equations numerically.

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