Find the differential equation whose general solution is…

Find the differential equation whose general solution is given by $y=e^{cx}$, where $c$ is an arbitrary constant.

Explanation of Method

Since there is a single arbitrary constant, one differentiation should suffice to eliminate it. Differentiate the solution, then substitute back to remove $c$ algebraically.

Step 1: Differentiate

From $y=e^{cx}$, the chain rule gives

\[\frac{dy}{dx}=c\,e^{cx}=c\,y.\]

Step 2: Eliminate $c$

From $y=e^{cx}$, taking logarithms gives $\ln y = cx$, so $c=\dfrac{1}{x}\ln y$. Substituting into $y'=cy$,

\[\frac{dy}{dx}=\frac{y\ln y}{x}.\]

Result

Final answer:

\[\boxed{\,x\,\dfrac{dy}{dx}=y\ln y\,}\]