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

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

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

A particular solution to a differential equation is given by $y = \frac{x}{3} + \frac{4}{3}x^{-2}$ for $x > 0$.

Calculate the value of $y$ when $x = 3$.

By using the substitution $y = vx$, solve the differential equation to find an expression for $v$ in terms of $x$ and hence find the general solution for $y$ in terms of $x$.

Solve the differential equation

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

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

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

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

and write its explicit form.

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

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

Determine the $x$-value at which the particular solution through $(1,-1)$ crosses the $x$-axis, given

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

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

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

Find the particular solution of the differential equation $\frac{dy}{dx} = \frac{x - 2y}{x}$ given that $y = 1$ when $x = 2$.

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

$\frac{\text{d}y}{\text{d}x} = \frac{x - 2y}{x}$

is asymptotic to a straight line, and determine the equation of this line.

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

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

shows positive slopes.