Find the series solution of the differential equation $y ^ { tt } - 7 x y ^ { t } = 0 \text { given the initial }$ conditions $y ( 0 ) = 0$ and $y ^ { t } ( 0 ) = 4$
A) $y = 4 \left( 1 + \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n + 1 } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) } \right)$
B)$y = 4 \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$
C)$y = 4 \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n + 1 } x ^ { 2 n + 1 } } { 2 ^ { n } ( 2 n + 1 ) ! }$
D)$y = 4 + \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } ( 2 n + 1 ) ! }$
E)$y = 4 + \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$

Question

Quizplus · Accepted Answer

The correct series solution is obtained by recognizing the pattern of coefficients that satisfy the differential equation and the initial conditions. The term $4$ in front of the summation accounts for the initial condition $y^t(0) = 4$, and the structure of the series inside the summation correctly reflects the iterative process of solving the differential equation with the given initial conditions. The coefficients in option B correctly match the pattern that would be derived from applying the method of series solutions to the differential equation, taking into account the initial conditions.

Find the Series Solution of the Differential Equation ytt−7xyt=0 given the initial y ^ { tt } - 7 x y ^ { t } = 0 \text { given the initial }ytt−7xyt=0 given the initial

Find the Series Solution of the Differential Equation $y ^ { tt } - 7 x y ^ { t } = 0 \text { given the initial }$