The recurrence relation for the differential equation $2 x y ^ { \prime \prime } - y ^ { t } + 2 y = 0$ is
A) $c _ { k + 1 } ( k + r ) ( 2 k + 2 r - 1 ) + 2 c _ { k } = 0$
B) $c _ { k + 1 } ( k + r ) ( k + r - 1 ) + 2 c _ { k } = 0$
C) $c _ { k + 1 } ( k + r + 1 ) ( 2 k + 2 r - 1 ) - 2 c _ { k } = 0$
D) $c _ { k + 1 } ( k + r + 1 ) ( 2 k + 2 r - 1 ) + 2 c _ { k } = 0$
E) $c _ { k + 1 } ( k + r + 1 ) ( 2 k + 2 r ) + 2 c _ { k } = 0$

Question

Quizplus · Accepted Answer

The given differential equation is $2xy'' - y' + 2y = 0$. When solving such equations using a power series method, we assume a solution of the form $y = \sum_{k=0}^{\infty} c_k x^{k+r}$, where $c_k$ are the coefficients to be determined and $r$ is the lowest power in the series. Differentiating $y$ and $y''$ and substituting into the differential equation, we aim to align terms to find a recurrence relation for the coefficients $c_k$. The correct recurrence relation that emerges from this process, considering the differential equation's structure and the operations involved in differentiating and substituting the series back into the equation, is given by option D: $c _ { k + 1 } ( k + r + 1 ) ( 2 k + 2 r - 1 ) + 2 c _ { k } = 0$. This option correctly accounts for the coefficients and powers involved in the series solution of the differential equation.

The Recurrence Relation for the Differential Equation 2xy′′−yt+2y=02 x y ^ { \prime \prime } - y ^ { t } + 2 y = 02xy′′−yt+2y=0

The Recurrence Relation for the Differential Equation $2 x y ^ { \prime \prime } - y ^ { t } + 2 y = 0$