Find value for $x$ for which $e^y \frac {dy}{dx} = 6$

Question

Find value for $x$ for which $$e^y \frac {dy}{dx} = 6$$

All help appreciated. All working please.

Thanks.

Answer 1

This is a rather simple differential equation to solve, as its already in separable form :

$$e^y \frac{\mathrm{d}y}{\mathrm{d}x}=6 \Rightarrow \int e^y\mathrm{d}y = \int6\mathrm{d}x \Leftrightarrow e^y= 6x + c \Leftrightarrow y(x) = \ln(6x+c)$$

Now, if you yield an initial condition $y(0) = y_0$, one would get for the constant $c$ :

$$y(0) = y_0 \Rightarrow \ln(c) = y_0 \Leftrightarrow c = e^{y_0}$$