Solve the differential equation.
-$$\frac { d y } { d x } = e ^ { 5 x - 5 y }$$
A) $y = 5 e ^ { 5 x } + C$
B) $y = 5 \ln \left( e ^ { 5 x } + C \right)$
C) $y = \ln \left( e ^ { 5 x } + C \right)$
D) $y = \frac { 1 } { 5 } \ln \left( e ^ { 5 x } + C \right)$

Question

Quizplus · Accepted Answer

To solve the differential equation $\frac { d y } { d x } = e ^ { 5 x - 5 y }$, we can rewrite it as $e^{5y} dy = e^{5x} dx$. Integrating both sides gives $\int e^{5y} dy = \int e^{5x} dx$, leading to $\frac{1}{5}e^{5y} = \frac{1}{5}e^{5x} + C$. Taking the natural logarithm of both sides results in $5y = \ln(e^{5x} + C)$, and dividing by 5 gives $y = \frac { 1 } { 5 } \ln \left( e ^ { 5 x } + C \right)$.

Solve the Differential Equation dydx=e5x−5y\frac { d y } { d x } = e ^ { 5 x - 5 y }dxdy​=e5x−5y

Solve the Differential Equation $\frac { d y } { d x } = e ^ { 5 x - 5 y }$