To solve the given problem, we are tasked with finding the first four nonzero terms in a power series expansion for the solution of the differential equation

$(x^2 + 18)y'' + y = 0$

about $x = 0$.

1. Power Series Assumption

Assume $y(x)$ is given as a power series: $y(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots$

The derivatives are: $y'(x) = \sum_{n=1}^\infty n a_n x^{n-1} = a_1 + 2a_2 x + 3a_3 x^2 + \cdots$ $y''(x) = \sum_{n=2}^\infty n(n-1)a_n x^{n-2} = 2a_2 + 6a_3 x + 12a_4 x^2 + \cdots$

2. Substitution into the Differential Equation

Substitute $y(x)$, $y'(x)$, and $y''(x)$ into the equation $(x^2 + 18)y'' + y = 0$: $(x^2 + 18)\left(\sum_{n=2}^\infty n(n-1)a_n x^{n-2}\right) + \sum_{n=0}^\infty a_n x^n = 0$

Expand $(x^2 + 18)y''$: $(x^2 + 18)y'' = x^2 \left(\sum_{n=2}^\infty n(n-1)a_n x^{n-2}\right) + 18\left(\sum_{n=2}^\infty n(n-1)a_n x^{n-2}\right)$ $= \sum_{n=2}^\infty n(n-1)a_n x^n + \sum_{n=2}^\infty 18n(n-1)a_n x^{n-2}$

Rewriting both terms with the $y$ series, we combine like powers of $x$.

3. Simplifying and Recurrence Relation

Matching coefficients of $x^n$, we find a recurrence relation: $n(n-1)a_n + 18(n+2)(n+1)a_{n+2} + a_n = 0$

For specific terms:

1. $n = 0$: $a_0 + 18a_2 = 0 \implies a_2 = -\frac{a_0}{18}$

2. $n = 1$: $a_1 + 18 \cdot 3 \cdot 2 a_3 = 0 \implies a_3 = -\frac{a_1}{108}$

3. $n = 2$: Solve for $a_4$ in terms of $a_0$, using the pattern established above.

4. First Four Nonzero Terms

Using the recurrence relations, the first four terms are: $y(x) = a_0 \left(1 - \frac{x^2}{18}\right) + a_1 \left(x - \frac{x^3}{108}\right)$

Thus, the solution is: $y(x) = a_0 - \frac{a_0}{18}x^2 + a_1 x - \frac{a_1}{108}x^3$

Would you like me to provide further details on deriving the recurrence relation or verifying the solution?

Relative Questions:

1. How does the recurrence relation help in solving power series expansions?
2. What are the conditions for a power series solution to converge?
3. How would the solution change if additional terms (higher order) were needed?
4. Why is it necessary to solve for terms in terms of $a_0$ and $a_1$?
5. How does the presence of the $x^2 + 18$ factor affect the recurrence relation?

Tip:

When solving differential equations using power series, carefully expand each term and match coefficients systematically to avoid algebraic