1. Find the general solution of $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2x-4$$ and sketch three particular solutions to this differential equation.
$y = x^2 - 4x + c$
2. Solve the differential equation $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = x^3y^2$$ giving your answer in the form $y = ...$
$y=\dfrac{4}{c-x^4}$
3. Find the particular solution to the differential equation $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{x}{y}$$ that passes through the point $(0,7)$.
$y^2 = 49-x^2$
4. Solve the differential equation $$\dfrac{\mathrm{d}y}{\mathrm{d}x} - y=5$$ in the form $y = ...$
$y = A\mathrm{e}^x-5$
5. Find the solution to the differential equation $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = x^2(1+y)$$ when $y = 3$ and $x = 0$. Write your answer in the form $y = ...$
$y =4\mathrm{e}^{\frac{1}{3}x^3}-1$
6. Find the solution to the differential equation $$y\dfrac{\mathrm{d}y}{\mathrm{d}x} -\mathrm{e}^x = 0$$ when $y = 4$ and $x = 0$. Write your answer in the form $y^2 = ...$
$y^2 = 2\mathrm{e}^x + 14$
7. Find the general solution of $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{3y+1}{2x}$$ in theform $y = ...$ and sketch three particular solutions to this differential equation.
$y = \dfrac{1}{3}\left(Ax^{1.5}-1\right)$
8. Find the general solutions to the following in the form $y = \mathrm{f}(x)$:
$\dfrac{\mathrm{d}y}{\mathrm{d}x} = (1+y)(1-2x)$
$\dfrac{\mathrm{d}y}{\mathrm{d}x} = y\tan x$
$y = A\mathrm{e}^{x-x^2}-1$
$y = A\sec x$
9. Find the general solution to the differential equation $$\cos^2x\dfrac{\mathrm{d}y}{\mathrm{d}x} = y^2\sin^2x$$ in the form $y = \mathrm{f}(x)$
$y = \dfrac{1}{x-\tan x + c}$
10. Find the general solution to the differential equation $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2\mathrm{e}^{x-y}$$ in the form $y = \mathrm{f}(x)$
$y = \ln|2\mathrm{e}^x + c|$
11. Find the particular solution to the differential equation $$x^2\dfrac{\mathrm{d}y}{\mathrm{d}x} = y$$ which satisfies the boundary condition $y = e^4$ at $x = -1$. Leave your answer in the form $y = ...$
$y = \mathrm{e}^{-\frac{1}{x}+3}$
12. Given that $x = 0$ when $y = 0$, find the particular solution to the differential equation $$(2y + 2yx)\dfrac{\mathrm{d}y}{\mathrm{d}x} = 1 - y^2$$ in the form $y^2=...$
$y^2 = \dfrac{x}{x+1}$
13. Find the particular solution to the differential equation $$(1-x^2)\dfrac{\mathrm{d}y}{\mathrm{d}x} = xy + y$$ with boundary conditions $y = 6$ at $x = 0.5$.
$y=\dfrac{3}{1-x}$
14. Given that $x\geqslant 3$, find the general solution to the differential equation $$(x-2)(3x-8)\dfrac{\mathrm{d}y}{\mathrm{d}x} = (8x-18)y$$ in the form $y = ...$
$y = A(x-2)(3x-8)^{\frac{5}{3}}$
15. Solve $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\sqrt{1+2y}}{x^2}$$ given that $y = 4$ when $x = 1$. Write your answer in the form $y = ...$
16. Solve $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6xy^2$$ given that $y = 1$ when $x = 2$. Write your answer in the form $y = ...$
$y = \dfrac{1}{13-3x^2}$
17. Solve $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{1}{40}(y-15)$$ given that $y = 85$ when $x = 0$. Write your answer in the form $y = ...$
$y = 15 + 70\mathrm{e}^{-\frac{x}{40}}$
18. Solve $$\dfrac{\mathrm{d}x}{\mathrm{d}t} = -kt\mathrm{e}^{\frac{1}{2}x}$$ where $k$ is a positive constant, given that $x = 6$ when $t = 0$. Write your answer in terms of $k$ and in the form $x = ...$
19. The height of a drone, $y$, over time, $t$, is modelled by the differential equation $$\dfrac{\mathrm{d}y}{\mathrm{d}t} = \dfrac{150\cos2t}{y}$$ The drone is 20 m high at $t = \dfrac{\pi}{4}$ seconds. Find the earliest time at which it will be 11 m high.
$2.09$
20. Find the particular solutions to $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = \sec^2x\sec^2y$$ given $y = 0$ when $x = \frac{\pi}{4}$. Write your answer in the form $4\tan x = ...$
$4\tan x = 4+2y+\sin 2y$
21. The proportion of a school unaffected by a virus, $p$%, $t$ days after the start of the school year is given by $$\dfrac{\mathrm{d}p}{\mathrm{d}t} = -\dfrac{1}{5}(p+1)^{0.5}$$ Initially 80% of the school are unaffected. Find the proportion of people unaffected after 60 days.
$8$%
22. A spherical water balloon is leaking and its volume, $V$ cm3, is reducing at a rate proportional to time, $t$ seconds. Initially, the radius of the balloon is 6 cm. 9 seconds later, the radius has halved. Find the time taken, in the form $a\sqrt{14}$ where $a$ is a constant, before there is no water left in the balloon.
$\dfrac{18}{7}\sqrt{14}$
23. The height of a tree, $y$ metres, increases over time, $t$ days, according to the differential equation $$\dfrac{\mathrm{d}y}{\mathrm{d}t} = \dfrac{1}{15y\sqrt{2y-1}}$$ Initially, the tree is 1 metre tall.
Solve the differential equation, leaving your answer in the form $t = ...$
Find the time taken, in days, for the tree to reach 2 metres.
$t=(3y+1)(2y-1)^{\frac{3}{2}}-4$
$32.4$
24. Solve $$\dfrac{\mathrm{d}y}{\mathrm{d}x} = y^2x\sin3x$$ given that $y = 1$ when $x = \dfrac{\pi}{6}$. Write your answer in the form $y = ...$
