Find a power series solution of the differential equation given below. Determine the radius of convergence of the resulting​ series, and use the series given below to identify the series in terms of familiar elementary functions. y prime minus 20 xyequals0 LOADING... Click the icon to view power series representations of elementary functions. Question content area bottom Part 1 The power series solution is ​y(x)equals    enter your response hereplusmidline ellipsis ​(Type an expression in terms of c 0 that includes all terms up to order​ 6.)

To solve the differential equation \( y' - 20xy = 0 \) using a power series, let's assume a solution of the form:

\[ y(x) = \sum_{n=0}^{\infty} a_n x^n \]

First, we differentiate this power series:

\[ y'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1} \]

Substituting \( y \) and \( y' \) into the differential equation \( y' - 20xy = 0 \), we get:

\[ \sum_{n=1}^{\infty} n a_n x^{n-1} - 20x \sum_{n=0}^{\infty} a_n x^n = 0 \]

Rewriting the second sum:

\[ 20x \sum_{n=0}^{\infty} a_n x^n = 20 \sum_{n=0}^{\infty} a_n x^{n+1} \]

Changing the index of summation in the second sum:

\[ 20 \sum_{n=1}^{\infty} a_{n-1} x^n \]

Now our equation becomes:

\[ \sum_{n=1}^{\infty} n a_n x^{n-1} - 20 \sum_{n=1}^{\infty} a_{n-1} x^n = 0 \]

To combine these sums, we can shift the index of the first sum up by 1:

\[ \sum_{n=1}^{\infty} n a_n x^{n-1} = \sum_{n=0}^{\infty} (n+1) a_{n+1} x^n \]

Now the equation is:

\[ \sum_{n=0}^{\infty} (n+1) a_{n+1} x^n - 20 \sum_{n=1}^{\infty} a_{n-1} x^n = 0 \]

This can be written as:

\[ (a_1) + \sum_{n=1}^{\infty} ((n+1) a_{n+1} - 20 a_{n-1}) x^n = 0 \]

For this equation to be true for all \( x \), the coefficients of each power of \( x \) must be zero. Thus, we get the following recurrence relation:

\[ (n+1) a_{n+1} = 20 a_{n-1} \]

Solving this recurrence relation, we start with \( n = 0 \):

\[ a_1 = 20 a_{-1} \]

Since there is no \( a_{-1} \), we set \( a_1 = 0 \). Now let's find \( a_2 \):

\[ (2) a_2 = 20 a_0 \implies a_2 = \frac{20}{2} a_0 = 10 a_0 \]

Next, for \( a_3 \):

\[ (3) a_3 = 20 a_1 \implies a_3 = \frac{20}{3} a_1 = 0 \]

And for \( a_4 \):

\[ (4) a_4 = 20 a_2 \implies a_4 = \frac{20}{4} a_2 = 5 a_2 = 50 a_0 \]

For \( a_5 \):

\[ (5) a_5 = 20 a_3 \implies a_5 = \frac{20}{5} a_3 = 0 \]

And for \( a_6 \):

\[ (6) a_6 = 20 a_4 \implies a_6 = \frac{20}{6} a_4 = \frac{10}{3} a_4 = \frac{10}{3} \cdot 50 a_0 = \frac{500}{3} a_0 \]

So, the power series up to order 6 is:

\[ y(x) = a_0 \left( 1 + 10 x^2 + 50 x^4 + \frac{500}{3} x^6 \right) + \cdots \]

Given that \( a_0 \) is arbitrary, let \( a_0 = c_0 \):

\[ y(x) = c_0 \left( 1 + 10 x^2 + 50 x^4 + \frac{500}{3} x^6 \right) + \cdots \]

To determine the radius of convergence, we can use the ratio test. For large \( n \):

\[ \left| \frac{a_{n+2}}{a_n} \right| = \left| \frac{20 a_n}{(n+2)(n+1)} \right| \]

The radius of convergence \( R \) is given by:

\[ \lim_{n \to \infty} \left| \frac{a_{n+2}}{a_n} \right|^{-\frac{1}{2}} = \infty \]

Hence, the radius of convergence is \( R = \infty \).

Finally, we recognize that the series represents the exponential function \( e^{10x^2} \), as:

\[ e^{10x^2} = \sum_{n=0}^{\infty} \frac{(10x^2)^n}{n!} \]

Thus, the solution to the differential equation \( y' - 20xy = 0 \) is:

\[ y(x) = c_0 e^{10x^2} \]

Would you like more details or have any questions? Here are some related questions:

1. How do you determine a power series solution for a different differential equation?
2. What are the steps to find the radius of convergence for a power series?
3. Can you provide another example of solving a differential equation using power series?
4. What is the importance of the ratio test in determining the radius of convergence?
5. How can you identify a power series in terms of familiar elementary functions?
6. What are some other methods to solve differential equations besides power series?
7. Can you explain the recurrence relation in more detail?
8. What if the initial condition changes, how would it affect the power series solution?

**Tip:** When dealing with power series, always check if the series can be recognized as a known elementary function to simplify the solution.

Explore the power series solution method for the differential equation y' - 20xy = 0. Learn how to determine the radius of convergence and identify the resulting series in terms of elementary functions. Suitable for advanced undergraduate students studying differential equations.

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