sketch the slope field for each differential equation.\n1) $\frac{dy}{dx}=x$\n2) $\frac{dy}{dx}=-\frac{x}{y}$

sketch the slope field for each differential equation.\n1) $\frac{dy}{dx}=x$\n2) $\frac{dy}{dx}=-\frac{x}{y}$

sketch the slope field for each differential equation.\n1) $\frac{dy}{dx}=x$\n2) $\frac{dy}{dx}=-\frac{x}{y}$

Answer

Explanation:

Step1: Determine slope at points for $\frac{dy}{dx}=x$

For different $x$ - values, find the slope. When $x = - 2$, $\frac{dy}{dx}=-2$; when $x=-1$, $\frac{dy}{dx}=-1$; when $x = 0$, $\frac{dy}{dx}=0$; when $x = 1$, $\frac{dy}{dx}=1$; when $x = 2$, $\frac{dy}{dx}=2$. At each point $(x,y)$ in the plane, draw a short line - segment with slope equal to the value of $\frac{dy}{dx}$ at that $x$ - value (the slope does not depend on $y$ in this case).

Step2: Determine slope at points for $\frac{dy}{dx}=-\frac{x}{y}$

For example, at the point $(1,1)$, $\frac{dy}{dx}=-1$; at the point $(1, - 1)$, $\frac{dy}{dx}=1$; at the point $(-1,1)$, $\frac{dy}{dx}=1$; at the point $(-1,-1)$, $\frac{dy}{dx}=-1$. At the points where $y = 0$, the slope is undefined. For each non - zero $y$ and corresponding $x$ value, draw a short line - segment with slope equal to $-\frac{x}{y}$.

Answer:

To sketch the slope fields:

  1. For $\frac{dy}{dx}=x$:
    • At $x=-2$, draw short line - segments with slope $-2$ at all $y$ values on the vertical line $x = - 2$.
    • At $x=-1$, draw short line - segments with slope $-1$ at all $y$ values on the vertical line $x=-1$.
    • At $x = 0$, draw short horizontal line - segments (slope $0$) at all $y$ values on the $y$ - axis.
    • At $x = 1$, draw short line - segments with slope $1$ at all $y$ values on the vertical line $x = 1$.
    • At $x = 2$, draw short line - segments with slope $2$ at all $y$ values on the vertical line $x = 2$.
  2. For $\frac{dy}{dx}=-\frac{x}{y}$:
    • At points like $(1,1)$ draw a short line - segment with slope $-1$.
    • At points like $(1, - 1)$ draw a short line - segment with slope $1$.
    • Avoid drawing at points on the $x$ - axis ($y = 0$) since the slope is undefined. Draw short line - segments with appropriate slopes at other points in the plane according to the formula $-\frac{x}{y}$.