Solve the differential equation $$ \frac{dy}{dt} = ky $$ for a constant growth rate $$ k $$.

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To solve the differential equation

$$
\frac{dy}{dt} = ky,
$$

we can use the method of separation of variables. We start by rearranging the equation to separate the variables $$y$$ and $$t$$:

$$
\frac{dy}{y} = k \, dt.
$$

Next, we integrate both sides. The left side integrates to $$\ln |y|$$, and the right side integrates to $$kt$$:

$$
\int \frac{dy}{y} = \int k \, dt \implies \ln |y| = kt + C,
$$

where $$C$$ is the constant of integration.

To solve for $$y$$, we exponentiate both sides:

$$
|y| = e^{kt + C} = e^{kt} e^C.
$$

Let $$C' = e^C$$, which is a positive constant. Thus, we can write:

$$
|y| = C' e^{kt}.
$$

Since $$C'$$ is positive, we can drop the absolute value by allowing $$C'$$ to take on any real value (positive or negative):

$$
y = C' e^{kt}.
$$

Finally, we can express the solution in a more general form by letting $$C = C'$$ (which can be any real number):

$$
y(t) = Ce^{kt},
$$

where $$C$$ is a constant determined by initial conditions. This is the general solution to the differential equation $$ \frac{dy}{dt} = ky $$.
The general solution to the differential equation $$ \frac{dy}{dt} = ky $$ is $$ y(t) = Ce^{kt} $$, where $$C$$ is a constant.

Solve the differential equation for a constant growth rate .

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