If \(y=\sum_{n=0}^{\infty}a_nx^n\), write \(y'\) as a power series.

If \(y=\sum_{n=0}^{\infty}a_nx^n\), write \(y''\) as a power series in powers of \(x^n\).

Find the recurrence relation for a power series solution of \(y'-y=0\) about \(x=0\), using \(y=\sum_{n=0}^{\infty}a_nx^n\).

Using the recurrence \(a_{n+1}=a_n/(n+1)\), express \(a_1\), \(a_2\), and \(a_3\) in terms of \(a_0\).

Find the first four non-zero terms of the power series solution to \(y'=y\) with \(y(0)=1\).

Find the recurrence relation for \(y''+y=0\), using \(y=\sum_{n=0}^{\infty}a_nx^n\).

For \(y''+y=0\) with \(y(0)=1\) and \(y'(0)=0\), use the recurrence to find terms through \(x^4\).

For \(y''+y=0\), explain why one power series solution contains only even powers when \(a_1=0\).

Find the first three non-zero terms of the solution to \(y''-y=0\) with \(y(0)=0\), \(y'(0)=1\).

For the differential equation \(xy''+y'=0\), substitute \(y=\sum_{n=0}^{\infty}a_nx^n\) and find the coefficient condition for powers \(x^n\) with \(n\ge0\).