Is $y = - \ln \left( e ^ { - x } - \frac { 15 } { 4 } \right)$ the solution to the differential equation $\frac { d y } { d x } = e ^ { y - x }$ with $y ( \ln 4 ) = - \ln 4$ ?

Question

Quizplus · Accepted Answer

To verify if $y = - \ln \left( e ^ { - x } - \frac { 15 } { 4 } \right)$ is a solution to the differential equation $\frac { d y } { d x } = e ^ { y - x }$, we differentiate $y$ with respect to $x$ and check if it satisfies the given differential equation. Differentiating $y$ gives $\frac { d y } { d x } = \frac { e ^ { - x } } { e ^ { - x } - \frac { 15 } { 4 } }$. This does not simplify to $e ^ { y - x }$ directly, indicating that the given function is not a solution to the differential equation as stated.

Is y=−ln⁡(e−x−154)y = - \ln \left( e ^ { - x } - \frac { 15 } { 4 } \right)y=−ln(e−x−415​)

Is $y = - \ln \left( e ^ { - x } - \frac { 15 } { 4 } \right)$