Solve the differential equation. Select the correct answer.
$$\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 4 y = 0$$
A)
$$y ( t ) = t e ^ { \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 15 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 15 } t } { 2 } \right) \right]$$
B)
$$y ( t ) = t ^ { 2 } e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \sqrt { \frac { 15 t } { 2 } } \right) + c _ { 2 } \sin \left( \sqrt { \frac { 15 t } { 2 } } \right) \right]$$
C)
$$y ( t ) = e ^ { \frac { \sqrt { t } } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { t } } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { t } } { 2 } \right) \right]$$
D)
$$y ( t ) = c e ^ { - \frac { t } { 2 } } \cos \left( \sqrt { \frac { 15 t } { 2 } } \right)$$
E)
$$y ( t ) = e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 15 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 15 } t } { 2 } \right) \right]$$

Question

Quizplus · Accepted Answer

The characteristic equation is $r^2+r+4=0$, which has roots $r=-\frac{1}{2}\pm\frac{\sqrt{15}}{2}i$. Thus, the general solution of the differential equation is $$y(t)=e^{-t/2}(c_1\cos(\frac{\sqrt{15}t}{2})+c_2\sin(\frac{\sqrt{15}t}{2})).$$ This matches with option E.

Solve the Differential Equation d2ydt2+dydt+4y=0\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 4 y = 0dt2d2y​+dtdy​+4y=0

Solve the Differential Equation $\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 4 y = 0$