Use power series to solve the differential equation. $$( x - 10 ) y ^ { \prime } + 2 y = 0$$
A) $$y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 10 ^ { n } }$$
B) $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { n + 1 } { 10 ^ { n } } x ^ { n }$
C) $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { ( n + 1 ) } { x ^ { n } }$
D) $$y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } ( n + 1 ) x ^ { n }$$
E) $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { n x ^ { n } } { 10 ^ { n } }$

Question

Quizplus · Accepted Answer

To solve the differential equation using power series, we assume that the solution has the form $$y ( x ) = \sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$$
The derivative of this series is $$y ^ { \prime } ( x ) = \sum _ { n = 1 } ^ { \infty } n c _ { n } x ^ { n - 1 }$$
Substituting these into the differential equation, we get
$$( x - 10 ) \sum _ { n = 1 } ^ { \infty } n c _ { n } x ^ { n - 1 } + 2 \sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n } = 0$$
Multiplying out the first sum and shifting indices, we get
$$\sum _ { n = 0 } ^ { \infty } ( n + 1 ) c _ { n + 1 } x ^ { n } - 10 \sum _ { n = 1 } ^ { \infty } n c _ { n } x ^ { n } + \sum _ { n = 0 } ^ { \infty } 2 c _ { n } x ^ { n } = 0$$
Setting the coefficient of each power of x to zero gives a recurrence relation for the coefficients:
\begin{align*}
& ( n + 1 ) c _ { n + 1 } - 10 n c _ { n } + 2 c _ { n } = 0 \
\Rightarrow\ & ( n + 1 ) c _ { n + 1 } = ( 10 n - 2 ) c _ { n } \
\Rightarrow\ & c _ { n + 1 } = \frac { 10 n - 2 } { n + 1 } c _ { n }
\end{align*}
We can easily compute the first few coefficients:
\begin{align*}
c _ { 0 } & = c _ { 0 } \
c _ { 1 } & = 8 c _ { 0 } \
c _ { 2 } & = \frac { 76 } { 2 } c _ { 1 } = 304 c _ { 0 } \
c _ { 3 } & = \frac { 298 } { 3 } c _ { 2 } = \frac { 90848 } { 3 } c _ { 0 } \
c _ { 4 } & = \frac { 1196 } { 4 } c _ { 3 } = 716288 c _ { 0 } \
c _ { 5 } & = \frac { 4782 } { 5 } c _ { 4 } = \frac { 2055635456 } { 5 } c _ { 0 }
\end{align*}
Therefore, the solution is
$$y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { n + 1 } { 10 ^ { n } } x ^ { n }$$

Use Power Series to Solve the Differential Equation (x−10)y′+2y=0( x - 10 ) y ^ { \prime } + 2 y = 0(x−10)y′+2y=0

Use Power Series to Solve the Differential Equation $( x - 10 ) y ^ { \prime } + 2 y = 0$