Separable differential equations and families of solutionsEdexcel International A Level Further Maths: Mind map

Exam questions on Separable differential equations and families of solutions

1. A curve $C$ passes through the point $(0,3)$ and satisfies the differential equation $\frac{dy}{dx}=2xy$, where $y>0$.
   Show that $C$ has a minimum point at $(0,3)$.2 marks
2. A drink cools in a room at a constant temperature of $20\,^\circ\text{C}$. At time $t$ minutes its temperature is $\theta\,^\circ\text{C}$, and the rate of decrease of $\theta$ is proportional to $\theta-20$, with constant of proportionality $k>0$. Initially $\theta=80$.
   After $5$ minutes the temperature is $50\,^\circ\text{C}$. Find the exact value of $k$.2 marks
3. A curve satisfies the differential equation $\frac{dy}{dx}=xe^{-y}$.
   Find the general solution, giving $e^y$ in terms of $x$.3 marks