Use the substitution $y=vx$ to show that $x\frac{dv}{dx}=v^2-v-2$.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt=\ln \frac{P}{P_0}\left(\frac{N-P_0}{N-P}\right)$.

Solve the differential equation given that $y=-1$ when $x=1$. Give your answer in the form $y=f\left(x\right)$.

Solve the differential equation and show that a general solution is $x^2+y^2=cx$ where $c$ is a positive constant.

Show that $t$ + 1 is an integrating factor for this differential equation.

Given that $y\left(1\right)=1$, use Euler’s method with step length $h$ = 0.25 to find an approximation for $y\left(2\right)$. Give your answer to two significant figures.

Show that the general solution of this differential equation is $P=A{\text{e}}^{kt}$, where $A\in \mathbb{R}$.

$\underset{t\to \mathrm{\infty }}{\text{lim}}P$

the solution of the differential equation, giving your answer in the form $P=f\left(t\right)$.

Sketch the isoclines for $\frac{\text{d}y}{\text{d}x}=4$.

Find the value of $t$ at which the amount of salt in the tank is decreasing most rapidly.

Sketch the graph of $y=f\left(x\right)$ for $1⩽x⩽1.4$ .

Hence, or otherwise, solve the differential equation

$$\frac{\text{d}y}{\text{d}x}=\frac{x^2+3xy+y^2}{x^2},x>0,$$

given that $y=1$ when $x=1$. Give your answer in the form $y=g(x)$.

Consider the differential equation

$$\frac{\text{d}y}{\text{d}x}=f\left(\frac{y}{x}\right),x>0.$$

Use the substitution $y=vx$ to show that the general solution of this differential equation is

$$\int \frac{\text{d}v}{f(v)-v}=\ln x+\text{ Constant.}$$

Hence solve this differential equation. Give the answer in the form $y=f(x)$.

Given that the initial population is 1000, $L=10000$  and $m=0.003$, find the number of years it will take for the population to triple.

Show that $\frac{\text{d}P}{\text{d}t}=\frac{m}{L}P\left(L-P\right)$, where $m\in \mathbb{R}$.

the number of years it will take for the population to triple.

On the same set of axes, sketch and label isoclines for $\frac{\text{d}y}{\text{d}x}=-1, 0$ and $1$, and clearly indicate the value of each $y$-intercept.

Hence or otherwise, explain why the point $(1, 1)$ is a local minimum.

Find the solution of the differential equation $\frac{\text{d}y}{\text{d}x}=x-y$, which passes through the point $(1, 1)$. Give your answer in the form $y=f\left(x\right)$.

Sketch the graph of $y=f\left(x\right)$ on the same set of axes as part (a)(i).

Use Euler’s method, with step length $h=0.1$, to find an approximate value of $y$ when $x=1.4$.

Deduce a similar expression for $v(T+k)$ in terms of $k$.

Deduce the set of values for $p$ such that there are two points on the curve $y=f\left(x\right)$ where $\frac{\text{d}y}{\text{d}x}=0$. Give a reason for your answer.

the number of years it will take for the population to triple.

Show that $\sqrt{x^2+1}$ is an integrating factor for this differential equation.

Find the percentage error when $y\left(2\right)$ is approximated by the final rounded value found in part (a). Give your answer to two significant figures.

Solve the differential equation, for , giving your answer in the form $y=f\left(x\right)$.

the population after 10 years

By solving an appropriate differential equation, show that the particle’s velocity at time $t$ is given by $v\left(t\right)=(1+v_0){\text{e}}^{-t}-1$.

Show that the $x$-coordinate(s) of the points on the curve $y=f\left(x\right)$ where $\frac{\text{d}y}{\text{d}x}=0$ satisfy the equation $x^{p-1}=\frac{1}{p}$.

By using the result to part (b) (i), show that $v\left(T-k\right)={\text{e}}^k-1$.

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

With reference to the curvature of your sketch in part (c)(iii), and without further calculation, explain whether you conjecture $f\left(1.4\right)$ will be less than, equal to, or greater than your answer in part (a).

Sketch the graph of $x$ versus $t$ for 0 ≤ $t$ ≤ 60 and hence find the maximum amount of salt in the tank and the value of $t$ at which this occurs.

Use the substitution $y=vx$ to show that $\int \frac{dv}{f\left(v\right)-v}=\ln x+C$ where $C$ is an arbitrary constant.

By solving an appropriate differential equation and using the result from part (b) (i), find an expression for $s_{\text{max}}$ in terms of $v_0$.

Solve the equation $\frac{\text{d}y}{\text{d}x}=1+\frac{y}{x}$ for $y\left(1\right)=1$.

Explain why the graph of $y=f\left(x\right)$ does not intersect the isocline $\frac{\text{d}y}{\text{d}x}=1$.

Find a prediction for the number of years it will take for the population of the world to double.

Hence, show that $v\left(T-k\right)+v\left(T+k\right)\ge 0$.

Use Euler’s method, with a step length of $0.1$, to find an approximate value of $y$ when $x=1.5$.

Solve the differential equation $\frac{dy}{dx}=\frac{\ln {\displaystyle }{\displaystyle 2}{\displaystyle x}}{x^2}-\frac{2y}{x}, x>0$, given that $y=4$ at $x=\frac{1}{2}$.

Give your answer in the form $y=f\left(x\right)$.

By substituting $Y=\frac{dy}{dt}$, show that $Y=B{\text{e}}^{2t}$ where $B$ is a constant.

Let the two values found in part (c)(ii) be ${\lambda }_1$ and ${\lambda }_2$.

Verify that $y=F{\text{e}}^{{\lambda }_{1}t}+G{\text{e}}^{{\lambda }_{2}t}$ is a solution to the differential equation in (c)(i),where $G$ is a constant.

Hence, by solving this differential equation, show that $x\left(t\right)=\frac{200-40{\text{e}}^{-\frac{t}{4}}\left(t+5\right)}{t+1}$.

Find the equation of the isocline corresponding to $\frac{\text{d}y}{\text{d}x}=k$, where $k\ne 0$, $k\in \mathbb{R}$.

The rate of change of the amount of salt leaving the tank is equal to $\frac{x}{t+1}$.

Find the amount of salt that left the tank during the first 60 minutes.

Solve the differential equation, giving your answer in the form $f\left(x, y\right)=0$.

Show that there are no points of inflexion on the graph of $y$ against $x$.

Express $m^2-2m+4$ in the form ${\left(m-a\right)}^2+b$ , where $a\text{, }b\in \mathbb{Z}$.

Solve the differential equation $\frac{\text{d}P}{\text{d}t}=\frac{m}{L}P\left(L-P\right)$, giving your answer in the form $P=g\left(t\right)$.

Show that $\frac{\text{d}x}{\text{d}t}=0.021x$.

Show that such an isocline can never be a normal to any of the family of curves that satisfy the differential equation.

Show that $1+x^2$ is an integrating factor for this differential equation.

By using the result from part (a) or otherwise, solve the differential equation and hence show that the curve has equation $y=x\left(\tan \left(\ln x\right)-1\right)$.

By solving the differential equation, show that $P=\frac{1200k}{\left(k-1200\right){\text{e}}^{-{\displaystyle \frac{t}{5}}}+1200}$.

Show that the time $T$ taken for the particle to reach $s_{\text{max}}$ satisfies the equation ${\text{e}}^T=1+v_0$.

The graph of $y$ against $x$ has a local maximum between $x=2$ and $x=3$. Determine the coordinates of this local maximum.

By solving the differential equation $\frac{dy}{dt}=y$, show that $y=A{\text{e}}^t$ where $A$ is a constant.

Show that $\frac{dx}{dt}-x=-A{\text{e}}^t$.

Solve the differential equation in part (a)(ii) to find $x$ as a function of $t$.

By differentiating $\frac{dy}{dt}=-x+y$ with respect to $t$, show that $\frac{d^{2}y}{d{t}^2}=2\frac{{\displaystyle dy}}{{\displaystyle dt}}$.

Hence find $y$ as a function of $t$.

Hence show that $x=-\frac{B}{2}{\text{e}}^{2t}+C$, where $C$ is a constant.

Show that $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.

Find the two values for $\lambda$ that satisfy $\frac{d^{2}y}{d{t}^2}-2\frac{dy}{dt}-3y=0$.