Math topic中文

Solving separable

Solving for the general solutions and particular solutions of differential equations of the form $\dfrac{dy}{dx}=\mathrm{f}(x)\,\mathrm{g}(y)$ , including reducing a given differential equation to this form by means of a given substitution.

Practice questions

1The points $(x,y)$ on a curve satisfy the equations $x = \tfrac{1}{2}t^2 - 2t$ and $\dfrac{dy}{dx} = \dfrac{1}{2t - 1}$ …
2In a large town, the number of people infected by a particular virus $t$ days after the virus was first discovered is $x$ …
3Consider the second order differential equation $\frac{d^{2}x}{dt^{2}} + a\frac{dx}{dt} = b,$ where $a$ and $b$ are no…
4A freshly brewed cup of tea, initially at a temperature of $85^\circ\text{C}$ , is left out to cool in a room with a room…
5In a large town, the number of people infected by a particular virus $t$ days after the virus was first discovered is $x$ …
6The variables $x$ and $y$ are related by the differential equation $\pi \dfrac{dy}{dx} + y(3 - \pi \tan x) = 0$ . Using t…
7In harvesting of renewable natural resources, it is desirable that policies are formulated to allow maximal harvest of t…
8Given that $z = e^{2x}\frac{dy}{dx}$ , express $\frac{dz}{dx}$ in terms of $x$ , $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$ …
9A family of curves is defined by the differential equation $xy\,\frac{dy}{dx} = x^{2} + y^{2}.$ By substituting $y = ux$ …
10In military exercises, parachutists jump from stationary helicopters and their motion is tracked by sensors tagged to th…
11The Stoke's Law states that when a spherical object of radius $R$ is falling at velocity $v$ in a fluid, the resistance…
12The variables $x$ and $y$ are related by the differential equation $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = 2\mathrm{e}^{-2x} .$ …
13A heated metal block at $160^\circ\text{C}$ is left to cool in a laboratory with an ambient temperature of $30^\circ\text{C}$ …
14By using the substitution $u = x - y$ , solve the differential equation $\dfrac{dy}{dx} + 4(x - y)^2\cos^2 x = \sin^2 x.$ …
15A chemical substance Uru starts to form when $N$ units of Vibranium react with $M$ units of Kryptonite, where $M < N$ . L…
16It is given that $2xy\dfrac{dy}{dx} = x^{2} - y^{2}$ where $x>0$ , $y<0$ . Using the substitution $u = \dfrac{y}{x}$ , show…
17Show, by using the substitution $w=\dfrac{y}{t^{2}}$ , that the general solution of the differential equation $t\dfrac{dw}{dt}=w\!\,^{2}t^{2}+2wt-2w$ …
18In a chemical reaction, a compound X is formed from two compounds Y and Z. The masses in grams of X, Y and Z present at…
19In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number o…
20A family of curves is given by the differential equation $x\dfrac{dy}{dx}-y=x(x-y)$ . By substituting $y=vx$ , show that t…
21In a particular chemical reaction, every 2 grams of $U$ and 1 gram of $V$ are combined and converted to form 3 grams of…
22[Cambridge] In a chemical reaction a compound $X$ is formed from a compound $Y$ . The masses in grams of $X$ and $Y$ …
23Glucose is a type of sugar in our blood and our body uses it for energy. Having low sugar in the blood for long periods…
24To prepare for the General Election, a political party studies the effectiveness of spreading information on Instantglam…
25Show that the differential equation $\frac{\text{d}y}{\text{d}x} - \frac{2y}{x} = x^3$ may be reduced by the substitutio…
26The S-I model is used to study the spread of infectious diseases across different scenarios. In this question, it is ass…
27Section A: Pure Mathematics [40 marks] (i) Prove that the substitution $u = x^2 + y^2$ reduces the differential equation…
28The variables $w$ , $x$ and $y$ are connected by the following differential equations: $e^{2x}\frac{dw}{dx}=w^{2}$ and…
29Given that $z = e^{2x}\frac{dy}{dx}$ , express $\frac{dz}{dx}$ in terms of $x$ , $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ .…
30An epidemiologist is studying the spread of a disease, dengue fever, which is spread by mosquitoes, in town $A$ . $P$ is…
31[Cambridge/9709/2009] An underground storage tank is being filled with liquid as shown in the diagram. Initially the…
32Find $\dfrac{d}{dx}e^{-x^{2}}$ . Hence find $\displaystyle \int x^{3}e^{-x^{2}}\,dx$ . By using the substitution $z = e^{x^{2}}y$ …
33Botanists are studying how excessive logging is affecting the population of forest trees in a country in South-East Asia…
34In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number o…
35A heated metal block at $160^\circ$ C is left to cool in a laboratory with an ambient temperature of $30^\circ$ C. The te…
36A wafer fabrication company uses the floating-zone method to purify polysilicon ingots, each having a uniform cross-sect…
37In the theory of learning, the amount of material memorised by a person, $y$ units, in time $t$ minutes satisfies the di…
38In a chemical reaction, compounds X and Y react to form a product. Let $x$ and $y$ denote the concentrations, in mol per…
39[Cambridge/9709/2004] A rectangular reservoir has a horizontal base of area 1000 $\text{m}^2$ . At time $t = 0$ , it i…
40Jim places an 8 m rod $XY$ against a vertical wall, as shown in the diagram below. The points $O, X$ and $Y$ are coplana…
41Using integration by parts, or otherwise, find $\displaystyle \int \frac{1}{y}\ln y\,dy$ . Given that $y>1$ and $\displaystyle \frac{dy}{dx}=\frac{y}{\ln y}$ …
42Baby Peter is pulling a toy wagon on a flat ground which is modelled by the $xy$ -coordinate plane. Peter is initially at…
43It is given that $2xy\,\dfrac{dy}{dx} = x^{2} - y^{2}$ where $x > 0$ , $y < 0$ . Using the substitution $u = \dfrac{y}{x}$ …
44A metal ball is released from the surface of the liquid in a tall cylinder. The ball falls vertically through the liquid…
45Using integration by parts, or otherwise, find $\displaystyle\int \frac{1}{y}\ln y\,dy$ . Given that $y>1$ and $\displaystyle\frac{dy}{dx}=\frac{y}{\ln y}$ …
46The molecular geometry is the three-dimensional arrangement of atoms in a molecule. Fig. 1 shows the molecular geometry…
47It is given that $2xy\frac{\text{d}y}{\text{d}x} = x^2 - y^2$ where $x > 0$ , $y < 0$ . Using the substitution $u = \frac{y}{x}$ …
48[Cambridge/H2 Maths 2007] The current $I$ in an electric circuit at time $t$ satisfies the differential equation $4\frac{\text{d}I}{\text{d}t} = 2 - 3I.$ …
49In a chemical reaction, a compound $X$ is formed from two compounds $Y$ and $Z$ . The masses in grams of $X$ , $Y$ and $Z$ …
50Jesse leans a 6 metre rod $XY$ in the position as shown below. At time $t$ seconds, the two ends $X$ and $Y$ are $x$ and…
51It is given that $\frac{\mathrm{d}w}{\mathrm{d}x} = (w - 1)(2 - w), \quad \text{for } 1 < w < 2. \quad \mathbf{(I)}$ I…
52In a chemical reaction, compounds X and Y react to form a product. Let x and y denote the concentrations, in mol per kil…
53The tangent at the point $(x, y)$ on the curve with equation $y = \mathrm{f}(x)$ passes through the point $(a, x)$ , wher…
54By using the substitution $x = a\tan\theta$ , show that $\displaystyle \int \frac{1}{x\sqrt{x^2 + a^2}}\,dx = \frac{1}{a}\ln\frac{x}{\sqrt{x^2 + a^2} + a} + c ,$ …
55A Red Lion skydiver jumps off from a helicopter. He leaves the helicopter with zero speed, and his speed $v\text{ ms}^{-1}$ …
56A team of naturalists is studying the change in population of wild boars on an island. It is suggested that the populati…
57In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number o…
58The diagram below shows the Gateway Arch, which is a monument in St. Louis, Missouri, United States. The arch stands at…
59Oil spill from a boat has a significant impact on the population of fish in a lake. The population (in thousands) $t$ da…