IB Mathematics Analysis and Approaches Topic AHL…

A population $P$ of bacteria grows according to the differential equation

$\frac{\mathrm{d} P}{\mathrm{~d} t}=\frac{P(1000-P)}{1000}, \quad t \geq 0$

(a) Solve the differential equation, given $P(0)=200$ , to find $P(t)$ .

(b) Find the exact time $t$ when the population reaches $500$ .

Consider the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{y}{x}=\sin x, \quad x>0

with the initial condition $y=0$ when $x=\frac{\pi}{2}$ .

(a) Find the general solution using the integrating factor method.

(b) the particular solution satisfying the initial condition.

Consider the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2 x+y}{x}, \quad x>0,

with the initial condition $y=0$ when $x=1$ .

(a) Solve the differential equation, giving your answer in the form $y=f(x)$ .

(b) Find the coordinates of the point where the solution curve intersects the line $y=x$ .

Consider the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 y}{x}=e^{x}, \quad x>0

with the initial condition $y=1$ when $x=1$ .

(a) Show that $x^{2}$ is an integrating factor for this differential equation.

(b) Solve the differential equation, giving your answer in the form $y=f(x)$ .

IB Mathematics Analysis and Approaches AHL 5.18—1st Order De’s – Euler Method, Variables Separable, Integrating Factor, Homogeneous DE Using Sub Y=vx Questionbank