Q. 59 - Calculus

Question

Sketch slope fields for each of the differential equations in Exercises $59-64$, and within each slope field sketch four different approximate solutions of the differential equation.

$\frac{dy}{dx}=-y$.

Step-by-Step Solution

Answer

A graph for the slope fields for the differential equation $\frac{dy}{dx}=-y$ is,

Step 1 . Given information

$\frac{dy}{dx}=-y$.

Step 2 . The slope field of a differential equation d y d x = g x , y comprises of the segments whose slope at any point a , b is given by, d y d x a , b .

In case the function $g\left(x,y\right)$ does not involve the independent variable $x$, then the slopes are same across each row of the slope field. In the present case, the differential equation $\frac{dy}{dx}=-y$ consists of the line segments whose slope at $\left(a,b\right)$ is equal to $-b$.

Step 3 . Draw the slope field of the differential equation given below.

Use the different colors to mark the different solutions.

Step 4 . Check the answer.

Note that the given differential equation is free from the variable $x$, so its solution can be readily obtained by antiderivative method as follows.

$$\begin{gathered} \frac{dy}{dx}=-y \\ \frac{dy}{y}=-dx \\ \int \frac{1}{y}dy=-\int dx \\ \ln \left|y\right|=-x+c \\ y=A{e}^{-x} \end{gathered}$$

Observe that the sketches drawn in the field are all exponential curves for $A=-2,1,2,4$. Hence, the approximate solutions are verified to be correct.