$\text { Which of the following is a solution of the differential equation } y ^ { ( 4 ) } - 10,000 y = 0 \text { ? }$
A) $y = 5 e ^ { - 10 x } + x \sin x$
B) $y = e ^ { - 10 x }$
C) $y = x ^ { 2 } \left( 10 + e ^ { x } \right)$
D) $y = - 10 x \cos x$
E) $y = 9 \ln x$

Question

$\text { Which of the following is a solution of the differential equation } y ^ { ( 4 ) } - 10,000 y = 0 \text { ? }$ 
A) $y = 5 e ^ { - 10 x } + x \sin x$
B) $y = e ^ { - 10 x }$
C) $y = x ^ { 2 } \left( 10 + e ^ { x } \right)$
D) $y = - 10 x \cos x$
E) $y = 9 \ln x$

Quizplus · Accepted Answer

The characteristic equation of the given differential equation is $r^4-10000=0$. We can factor this as $(r^2-100)(r^2+100)=0$, which gives the roots $r=\pm10i$ and $r=\pm100$. 
Therefore, the general solution is of the form $y(x)=c_1e^{100x}+c_2e^{-100x}+c_3\cos(10x)+c_4\sin(10x)$. 
Option A and C are not solutions to the given differential equation. 
Option D is a solution to the homogeneous equation $y^{(4)}-10000y=0$ but not to the non-homogeneous one. 
Option E is not even a polynomial, let alone a solution to the differential equation.

Therefore, the correct answer is B.

Which of the following is a solution of the differential equation y(4)−10,000y=0 ? \text { Which of the following is a solution of the differential equation } y ^ { ( 4 ) } - 10,000 y = 0 \text { ? } Which of the following is a solution of the differential equation y(4)−10,000y=0 ?

$\text { Which of the following is a solution of the differential equation } y ^ { ( 4 ) } - 10,000 y = 0 \text { ? }$