Solve the differential equation dy/dx = ky, where k is a constant.

Step-by-step solution

1. 1

   Recognise the model. This equation says the rate of change is proportional to the current amount — the law behind population growth, radioactive decay, continuously compounded interest and Newtonian cooling. Solving it once solves all of them.

2. 2

   Separate the variables. Dividing by $y$ requires $y \ne 0$, which we note and return to:

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

3. 3

   Integrate both sides.

   $\int \frac{1}{y}\,dy = \int k\,dx \quad\Longrightarrow\quad \ln|y| = kx + C_0$

   The absolute value is required because $\tfrac{1}{y}$ has antiderivative $\ln|y|$ on either side of zero.

4. 4

   Exponentiate and absorb the constants.

   $|y| = e^{kx + C_0} = e^{C_0}e^{kx}$

   Writing $A = e^{C_0} > 0$ and letting the sign of $y$ be carried by allowing $C = \pm A$, this becomes

   $y = Ce^{kx}$

5. 5

   Recover the excluded solution. The division by $y$ ruled out $y \equiv 0$, but that constant function does satisfy the equation — and it is recovered by $C = 0$. So allowing $C$ to be any real number makes $y = Ce^{kx}$ the complete general solution with nothing lost.

6. 6

   Verify and read off the behaviour. Differentiating, $y' = Cke^{kx} = k\left(Ce^{kx}\right) = ky$. With an initial condition $y(0) = y_0$ we get $C = y_0$, so $y = y_0e^{kx}$: growth when $k > 0$, decay when $k < 0$, and a constant when $k = 0$. The doubling time for $k > 0$ is $\tfrac{\ln 2}{k}$.