IB Mathematics Applications & Interpretation Question Type 1: Completing or Creating Slope Field Diagrams for Different Differential Equations Exercises

Determine the sign of $\frac{dy}{dx}$ 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.