QuestionDetermine the growth constant $k$, then find all solutions of the given differential equation. $$\begin{array}{l} y^{\prime}=0.19 y \\ k=\square \end{array}$$

Studdy Solution
Solve for $y$ by exponentiating both sides to eliminate the natural logarithm:
$$|y| = e^{0.19t + C}$$
This can be rewritten as:
$$y = \pm e^C \cdot e^{0.19t}$$
Let $A = \pm e^C$, where $A$ is an arbitrary constant. Thus, the general solution is:
$$y = A e^{0.19t}$$
The growth constant $k$ is:
$$\boxed{0.19}$$
The general solution to the differential equation is:
$$y = A e^{0.19t}$$
where $A$ is an arbitrary constant.