Solve the differential equation using the method of undetermined coefficients. $$y ^ { tt } - 5 y ^ { t } = \sin 15 x$$
A) $y ( x ) = c _ { 1 } e ^ { 5 x } + \frac { 1 } { 250 } \cos ( 6 x ) + \frac { 1 } { 750 } \sin ( 6 x )$
B)$y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 5 x } + \frac { 1 } { 750 } \cos ( 6 x ) - \frac { 1 } { 250 } \sin ( 6 x )$
C) $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { 5 x } - \frac { 1 } { 250 } \sin ( 6 x )$
D)$y ( x ) = c _ { 1 } e ^ { 5 x } + \frac { 1 } { 750 } \cos ( 6 x ) - \frac { 1 } { 250 } \sin ( 6 x )$
E)$y ( x ) = \frac { 1 } { 750 } \cos ( 6 x ) - \frac { 1 } { 250 } \sin ( 6 x )$

Question

Quizplus · Accepted Answer

The differential equation $y^{tt} - 5y^t = \sin(15x)$ is a second-order linear non-homogeneous differential equation. The method of undetermined coefficients suggests that the particular solution to the non-homogeneous part ($\sin(15x)$) is incorrect in all options, as they all mistakenly use $\sin(6x)$ or $\cos(6x)$ in their expressions. However, the homogeneous solution $y_h = c_1 + c_2e^{5x}$ is correctly identified in option B, which also includes arbitrary coefficients for a general solution. Despite the error in addressing the non-homogeneous part, B is the closest to a correct form given the options, assuming a typographical error in the sine function's argument.

Solve the Differential Equation Using the Method of Undetermined Coefficients ytt−5yt=sin⁡15xy ^ { tt } - 5 y ^ { t } = \sin 15 xytt−5yt=sin15x

Solve the Differential Equation Using the Method of Undetermined Coefficients $y ^ { tt } - 5 y ^ { t } = \sin 15 x$