Question

$The solution of the differential equation \frac{d y}{d x} = e^{x - y} i s$

$e^{x} + e^{- y} + k = 0$
$e^{2 x} = k e^{y}$
$e^{x} - e^{y} = k$
$e^{x + y} = k$

Answer: $e^{x} - e^{y} = k$

Solution :-

We have \frac{d y}{d x} = e^{x - y}

Now, use separation of variable method, we get

\begin{aligned} \frac{d y}{d x} & = & e^{x} \cdot e^{- y} \\ ⟹ e^{y} \cdot d y & = & e^{x} \cdot d x \dots \dots (1) \end{aligned}

Integrating both sides of eq(1), we get

\begin{aligned} \int e^{y} \cdot d y & = & \int e^{x} \cdot d x \\ e^{y} & = & e^{x} + k_{1} \\ e^{x} - e^{y} & = & - k_{1} \\ e^{x} - e^{y} & = & k \cdot (where, k = - k_{1}) \end{aligned}

So, correct answer is option-3, $\to e^{x} - e^{y} = k$

#Ordinary Differential Equation #BSEB PYQ 2017 #Mathematics #BSEB PYQs