Math / Calculus

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

STEP 1

1. The differential equation given is a first-order linear differential equation.
2. The equation is of the form $y' = ky$ , which is a standard form for exponential growth or decay.
3. The growth constant $k$ can be directly identified from the equation.

STEP 2

1. Identify the growth constant $k$ .
2. Solve the differential equation for $y$ .

STEP 3

Identify the growth constant $k$ from the differential equation. The equation given is:
$y' = 0.19y$
In this form, the growth constant $k$ is the coefficient of $y$ . Therefore:
$k = 0.19$

STEP 4

To solve the differential equation $y' = 0.19y$ , recognize that it is a separable differential equation. We can separate variables and integrate:
$\frac{dy}{y} = 0.19 \, dt$
Integrate both sides:
$\int \frac{1}{y} \, dy = \int 0.19 \, dt$
This gives:
$\ln |y| = 0.19t + C$
where $C$ is the constant of integration.

STEP 5

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.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms