$\text { Which of the following is a solution of the differential equation } x y ^ { t } - 4 y = x ^ { 5 } e ^ { x } \text { ? }$
A)$y = 4 x ^ { 5 } e ^ { 2 x }$
B) $y = 6 e ^ { 2 x } - 7 \sin 2 x$
C) $y = x ^ { 4 } e ^ { x }$
D) $y = 5 e ^ { - 2 x }$
E) $y = \ln x$

Question

$\text { Which of the following is a solution of the differential equation } x y ^ { t } - 4 y = x ^ { 5 } e ^ { x } \text { ? }$ 
A)$y = 4 x ^ { 5 } e ^ { 2 x }$
B) $y = 6 e ^ { 2 x } - 7 \sin 2 x$
C) $y = x ^ { 4 } e ^ { x }$
D) $y = 5 e ^ { - 2 x }$
E) $y = \ln x$

Quizplus · Accepted Answer

First, we need to confirm that the given function is a solution of the differential equation. We can do this by plugging in the given function in place of y and verifying that the equation holds true.

For choice A, 
$x(4x^5e^{2x}) - 4(4x^5e^{2x}) = x^5e^x$
$4x^6e^{2x} - 16x^5e^{2x} = x^5e^x$
This equation does not hold true, so it is not a solution.

For choice B, 
$x(6e^{2x}-7\sin2x) - 4(6e^{2x}-7\sin2x) = x^5e^x$
$6xe^{2x} - 7x\sin2x - 24e^{2x} + 28\sin2x = x^5e^x$
This equation also does not hold true, so it is not a solution.

For choice C, 
$x(x^4e^{x}) - 4(x^4e^{x}) = x^5e^x$
$x^5e^x - x^5e^x = x^5e^x$
This equation holds true, so choice C is a solution.

For choice D, 
$x(5e^{-2x}) - 4(5e^{-2x}) = x^5e^x$
$5xe^{-2x} - 20e^{-2x} = x^5e^x$
This equation does not hold true, so it is not a solution.

For choice E, 
$x(\ln x) - 4(\ln x) 
eq x^5e^x$
This equation obviously does not hold true, so it is not a solution.

Therefore, the only choice that is a solution of the differential equation is C.

Which of the following is a solution of the differential equation xyt−4y=x5ex ? \text { Which of the following is a solution of the differential equation } x y ^ { t } - 4 y = x ^ { 5 } e ^ { x } \text { ? } Which of the following is a solution of the differential equation xyt−4y=x5ex ?

$\text { Which of the following is a solution of the differential equation } x y ^ { t } - 4 y = x ^ { 5 } e ^ { x } \text { ? }$