First-order differential equations and integrating factorsEdexcel A-Level Further Maths: Mind map

Exam questions on First-order differential equations and integrating factors

1. Consider the differential equation $\frac{dy}{dx}+\frac{2}{x}y=x^2$ for $x>0$.
   Given that $y=1$ when $x=1$, find $y$ in terms of $x$.2 marks
2. A curve satisfies $\frac{dy}{dx}+y\tan x=\cos x$ for $-\frac{\pi}{2}<x<\frac{\pi}{2}$.
   The curve passes through the point $\left(\frac{\pi}{3},\frac{\pi}{3}\right)$. Find $y$ in terms of $x$.2 marks
3. A tank holds 200 litres of well-stirred brine containing 10 kg of salt at time $t=0$, where $t$ is in minutes. Brine of concentration 0.5 kg per litre flows in at 4 litres per minute, and the well-stirred mixture flows out at 4 litres per minute. Let $m$ kg be the mass of salt in the tank at time $t$.
   Show that $\frac{dm}{dt}+\frac{m}{50}=2$.3 marks