# Power Series - Differential Equation (check my answer)

• vucollegeguy
In summary, using the power series method, the solution to the differential equation y'+xy=0 when y(0)=1 is represented by the power series \sum(-1)^k \frac{x^{k+2}}{k!}, which can be recognized as the function cos(\frac{x}{2}).
vucollegeguy
Using the power series method to solve the differential equation
y'+xy=0 when y(0)=1

Write the solution in the form of a power series and then recognize what function it represents.
************************************

$$\sum$$(-1)k*[(x2k)/(2k)*k!]

Is it written in the form of a power series?
What function does it represent?

Thanks to anyone that helps!

Yes, it's correct. And yes, it's a power series. It looks like exp(f(x)) to me. Can you guess what f(x) might be?

I'd guess and say that f(x)=cos(x).
So the function would be f(x)=x2cos(x).

Please correct me if I am wrong.
Thank you for helping thus far.

Not quite. Your sum reduces to
$$\sum(-1)^k \frac{\left(\frac{x^2}{2}\right)^2}{k!}$$
which is a "cosine" but it is
$$cos(\frac{x}{2})$$

$$x^2cos(x)$$
would be
$$x^2\sum(-1)^k \frac{x^k}{k!}= \sum (-1)^k \frac{x^{k+2}}{k!}$$

Ok -got it.
A big "THANKKK YOUUU!" to all for all of your help!

## 1. What is a power series?

A power series is an infinite series of the form $\sum_{n=0}^{\infty} a_nx^n$, where $a_n$ are constants and $x$ is a variable. It is a useful tool in calculus and is often used to represent functions as an infinite sum of polynomials.

## 2. How are power series used in differential equations?

Power series can be used to find solutions to differential equations. By substituting the series into the equation and equating coefficients of like powers, a system of equations can be formed to find the coefficients. This allows for the solution of more complicated differential equations.

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

A Taylor series is a type of power series that represents a function as an infinite sum of its derivatives evaluated at a certain point. Power series can be used to find the coefficients of a Taylor series, allowing for the approximation of a function by a polynomial.

## 4. Can all functions be represented as a power series?

No, not all functions can be represented as a power series. In order for a function to be represented as a power series, it must be analytic, meaning it has derivatives of all orders. This condition is not met for all functions, such as the absolute value function.

## 5. How can power series be used to solve initial value problems?

Power series can be used to solve initial value problems by finding the coefficients of the series that satisfy the initial conditions. This results in a power series that represents the solution to the differential equation. By truncating the series at a certain point, an approximate solution can be found.

• Calculus and Beyond Homework Help
Replies
1
Views
449
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
590
• Calculus and Beyond Homework Help
Replies
17
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
543
• Calculus and Beyond Homework Help
Replies
3
Views
567
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
622
• Calculus and Beyond Homework Help
Replies
3
Views
708
• Calculus and Beyond Homework Help
Replies
22
Views
3K