Solve the differential equation dy/dx = y.

Step-by-step solution

1. 1

   Recognise the equation and what it says. The rate of change equals the quantity itself — the defining property of exponential growth. It is separable, since the right side depends on $y$ alone. Note immediately that $y=0$ is a constant solution, since then both sides are $0$; we will need to recover it at the end.

2. 2

   Separate the variables. Assuming $y\neq0$ for the moment, divide by $y$ and multiply by $dx$:

   $\frac{1}{y}\,dy=dx$

3. 3

   Integrate both sides.

   $\int\frac{1}{y}\,dy=\int1\,dx\qquad\Longrightarrow\qquad \ln|y|=x+C$

   The absolute value is required because $\ln$ of a negative number is undefined — and it is what allows negative solutions to survive.

4. 4

   Exponentiate to free y.

   $|y|=e^{x+C}=e^{C}e^{x}$

   Here $e^C$ is an arbitrary positive constant.

5. 5

   Absorb the sign into the constant. Removing the absolute value introduces a $\pm$: $y=\pm e^{C}e^{x}$. Since $\pm e^{C}$ ranges over all nonzero reals, rename it $C$:

   $y=Ce^{x}$

   Allowing $C=0$ now also restores the constant solution $y=0$ that step 2 had excluded, so this single formula is the complete general solution.

6. 6

   Verify by differentiation. If $y=Ce^x$ then $\dfrac{dy}{dx}=Ce^x=y$ ✓ — the equation is satisfied for every $C$. Checking numerically with $C=1.7$ at $x=-1,\,0.3,\,2.2$, the symmetric difference quotient matches $y$ itself to seven digits ✓.