Question Type 1: Completing or creating slope field diagrams for different differential equations Exercises

Calculate the slopes at the points $(1,1)$, $(2,0)$ and $(3,2)$ in the slope field for the differential equation

rac{dy}{dx}=rac{x-2y}{x}.

Determine the sign of $\displaystyle\frac{dy}{dx}$ at the point $(2,3)$ and state whether solution curves are increasing or decreasing there, given

rac{dy}{dx}=rac{x-2y}{x}.

For $x=1$, identify the interval(s) of $y$ for which the slope field defined by

rac{dy}{dx}=rac{x-2y}{x}

shows positive slopes.

Find all points on the line $x=1$ where the slope in the slope field is zero for

rac{dy}{dx}=rac{x-2y}{x}.

Using the particular solution found for $y(2)=1$, calculate the value of $y$ when $x=3$.

Solve the differential equation

rac{dy}{dx}=rac{x-2y}{x}

to find the general solution $y(x)$.

Find the particular solution of

rac{dy}{dx}=rac{x-2y}{x}

that satisfies the initial condition $y(2)=1$.

Determine the particular solution passing through $(1,0)$ for

$\frac{dy}{dx}=\frac{x-2y}{x}\,,$

and write its explicit form.

Describe the behavior of the particular solution through $(1,0)$ as $x\to0^+$.

Determine the $x$–value at which the particular solution through $(1,-1)$ crosses $y=0$ again, given

$\frac{dy}{dx}=\frac{x-2y}{x}\,.$

Show that as $x\to\infty$, the general solution of

$\frac{dy}{dx}=\frac{x-2y}{x}$

is asymptotic to a straight line, and determine its equation.

Solve the differential equation by the substitution $y=v x$, find $v(x)$, and verify that your result matches the general solution.