Solve the differential equation.
-$\frac { d y } { d x } = e ^ { 8 x - 8 y }$
A) $y = 8 e ^ { 8 x } + C$
B) $y = \ln \left( e ^ { 8 x } + C \right)$
C) $y = \frac { 1 } { 8 } \ln \left( e ^ { 8 x } + C \right)$
D) $y = 8 \ln \left( e ^ { 8 x } + C \right)$

Question

Quizplus · Accepted Answer

First, we need to separate the variables:
$$\frac { d y } { d x } = e ^ { 8 x - 8 y }$$
$$\frac { d y } { e ^ { -8 y } } = e ^ { 8 x } dx$$
Integrating both sides:
$$-\frac { 1 } { 8 } e ^ { -8 y } = \frac { 1 } { 8 } e ^ { 8 x } + C$$
$$e ^ { -8 y } = -e ^ { 8 x } - 8C$$
$$e ^ { 8 y } = \frac { 1 } { -e ^ { 8 x } - 8C }$$
$$y = \frac { 1 } { 8 } \ln \left( e ^ { 8 x } + C \right)$$
Therefore, the correct answer is option C.

Solve the Differential Equation dydx=e8x−8y\frac { d y } { d x } = e ^ { 8 x - 8 y }dxdy​=e8x−8y

Solve the Differential Equation $\frac { d y } { d x } = e ^ { 8 x - 8 y }$