Q. 60 - Calculus

Question

For each of the differential equations , draw slope fields and four distinct approximations of the differential equation's solutions inside each slope field.

$\frac{dy}{dx}=1-x$

Step-by-Step Solution

Answer

The slope fields and four distinct approximations of the differential equation is shown below,

Step 1: Given information

The differential equation $\frac{dy}{dx}=1-x$

Step 2: Explanation

Sketch four further approximations of the differential equation's slope field solutions.

A differential equation's $\frac{dy}{dx}=g(x,y)$  slope field consists of line segments whose slope at each point $(a,b)$ is provided by the expression ${\left.\frac{dy}{dx}\right|}_{(a,b)}.$The slopes in each column of the slope field are the same if the dependent variable $y$ is not involved in the function$g(x, y).$ The line segments that make up the differential equation $\frac{dy}{dx}=1-x$ in this instance have slopes that are equal to $1-a$ at $(a, b).$

The differential equation is presented below. Draw the slope field, then by observing the slopes' trend, draw four sketches to depict the differential equation's four approximative solutions. To identify the various solutions, use different colours.

Step 3: Calculation

Due to the fact that the differential equation $\left(1\right)$ is not dependent on the variable $y$, the following antidifferentiation method can be used to quickly find the solution.

$$\begin{gathered} \int dy=\int (1-x)dx \\ y=x-\frac{1}{2}{x}^2+C \end{gathered}$$

Remember that for various values of the constant $C,$ the aforementioned equation represents an inverted parabola (with vertex downwards). Observe that all of the sketches for $C=-3, 0, 2, 3$ are parabolic curves in the slope field. Thus, the accuracy of the approximations is confirmed.