Q. 12 - Calculus

Question

Explain in your own words how the slopes of the line segments in a slope field for a differential equation are related to the differential equation. 

Step-by-Step Solution

Answer

Ans: It is proved that the slopes of the line segments in a slope field for a differential equation are related to the differential equation.  

Step 1. Given information.

given,

      the slopes of the line segments in a slope field for a differential equation are related to the differential equation.  

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

In case the function $g(x,y)$ does not involve the dependent variable $y$, then the slopes are the same across each column of the slope field. For example, the slope field for the differential equation $\frac{dy}{dx}=x^2$ consists of line segments whose slope at $(a,b)$ is equal to $a^2$.

Step 3. And,

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