# Solving an ODE with power series

• I
• Kaguro
In summary, the conversation discusses solving an ODE using the regular power series method. After finding the derivatives and putting them in the ODE, a recurrence relation is obtained, which relates three coefficients. However, the conversation raises a question about how to proceed when there are multiple powers of x in each sum. The expert suggests transforming the index to get x^i in each sum and combining coefficients of "like powers" to solve the ODE.
Kaguro
TL;DR Summary
In power series method, after putting in the summations in the DE and simplifying, I found three powers of x.. how do I solve it?
I have an ODE:
(x-1)y'' + (3x-1)y' + y = 0
I need to find the solution about x=0. Since this is an ordinary point, I can use the regular power series solution.

Let y = ## \sum_{r=0}^\infty a_r x^r ##
after finding the derivatives and putting in the ODE, I have:
## \sum_{r=0}^\infty a_r (r)(r-1) x^{r-1} - \sum_{r=0}^\infty a_r (r)(r-1) x^{r-2} + 3 \sum_{r=0}^\infty a_r (r) x^{r} - \sum_{r=0}^\infty a_r (r) x^{r-1} + \sum_{r=0}^\infty a_r x^{r}=0 ##

Now I have three powers of x, so If I transform the index I'll still end up with a recurrence relation relating three coefficients.
I don't know how to proceed in that case.

Last edited:
When you put the power series into the ODE, what happened to the " = 0" part? It is necessary to use that to solve for the coefficients.

For each power of x, the total on the left must equal that power of x on the right, which is zero. Work on that and see how far it gets you.

Kaguro said:
Summary:: In power series method, after putting in the summations in the DE and simplifying, I found three powers of x.. how do I solve it?

I have an ODE:
(x-1)y'' + (3x-1)y' + y = 0
I need to find the solution about x=0. Since this is an ordinary point, I can use the regular power series solution.

Let y = ## \sum_{r=0}^\infty a_r x^r ##
after finding the derivatives and putting in the ODE, I have:
## \sum_{r=0}^\infty a_r (r)(r-1) x^{r-1} - \sum_{r=0}^\infty a_r (r)(r-1) x^{r-2} + 3 \sum_{r=0}^\infty a_r (r) x^{r} - \sum_{r=0}^\infty a_r (r) x^{r-1} + \sum_{r=0}^\infty a_r x^{r}=0 ##

Now I have three powers of x, so If I transform the index I'll still end up with a recurrence relation relating three coefficients.
I don't know how to proceed in that case.
Actually you have a many powers of x hidden in each sum. Also, though all your sums start at r= 0, in the first sum you have coefficients r and r- 1 so the first two terms are 0, It would be better to write it as $\sum_{r= 2}^\infty a_r r(r-1)x^{r- 2}$. In order to get $x^i$, let i= r- 2 so that r= i+ 2 and the sum becomes $\sum_{i= 0}^\infty a_{i+2}(i+2)(i+1)x^{i}$. Do the same with each of the other sums so that you have $x^i$ in each sum and can combine coefficients of "like powers".

## 1. What is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model many physical, biological, and chemical systems.

## 2. What is a power series?

A power series is an infinite series of the form ∑n=0^∞ cn(x-a)n, where cn are constants and a is a fixed point. It is a useful tool for approximating functions and solving differential equations.

## 3. How do you solve an ODE with power series?

To solve an ODE with power series, we first substitute the power series into the ODE and equate coefficients of like powers of x. This results in a system of equations that can be solved for the coefficients. The solution is then obtained by plugging the coefficients back into the power series.

## 4. What are the advantages of using power series to solve ODEs?

Power series provide a systematic and efficient method for solving ODEs. They allow for an exact solution in the form of an infinite series, which can be truncated to obtain a good approximation. They also allow for the study of the behavior of the solution near singular points.

## 5. Are there any limitations to using power series to solve ODEs?

One limitation of using power series to solve ODEs is that they may not converge for all values of x. This can be overcome by using other methods, such as numerical methods, to approximate the solution. Additionally, power series may not be able to capture all features of the solution, such as discontinuities or singularities.

• Differential Equations
Replies
4
Views
2K
• Differential Equations
Replies
1
Views
1K
• Differential Equations
Replies
1
Views
2K
• Differential Equations
Replies
7
Views
388
• Differential Equations
Replies
1
Views
1K
• Differential Equations
Replies
1
Views
1K
• Differential Equations
Replies
3
Views
1K
• Differential Equations
Replies
7
Views
1K
• Differential Equations
Replies
8
Views
524
• Calculus and Beyond Homework Help
Replies
2
Views
711