Differential Equations: exam review Exam review

The number \(N\) of students who have heard a rumour in a school of 800 students changes with time \(t\) days. The rate of increase of \(N\) is proportional to the product of the number of students who have heard the rumour and the number who have not. [5]

1. (a) (a) Write down a differential equation for \(N\) in terms of \(t\) and a constant \(k\). [2]
2. (b) (b) A curve satisfies \(\frac{dy}{dx}=6 x - \frac{4}{x^{2}}\) and passes through the point \((1,3)\). Find \(y\) in terms of \(x\). [3]

A chemical reaction is modelled by a differential equation for \(y\), the concentration in mol per litre, after \(x\) minutes. [6]

1. (a) (a) Find the general solution of \(\frac{dy}{dx}=xe^{-y}\), giving your answer in the form \(e^{y}=f(x)\). [3]
2. (b) (b) Given that \(y=1\) when \(x=0\), find \(y\) in terms of \(x\). [2]
3. (c) (c) Find the value of \(y\) when \(x=3\). [1]

A patient is given a drug. The mass \(m\) mg of the drug in the blood \(t\) hours after the dose satisfies \(\frac{dm}{dt}=-km\), where \(k\) is a positive constant. At \(t=0\) the mass is 250 mg and it is decreasing at 30 mg per hour. [7]

1. (a) (a) Find the value of \(k\). [2]
2. (b) (b) Solve the differential equation to show that \(m=250 e^{-kt}\). [2]
3. (c) (c) Find the half-life. [2]
4. (d) (d) Find the time taken for the amount to fall to 20 mg. [1]

A cup of tea at \(90^\circ\)C is left in a room at \(20^\circ\)C. Its temperature \(T^\circ\)C after \(t\) minutes satisfies \(\frac{dT}{dt}=-k(T-20)\), where \(k\) is a positive constant. [5]

1. (a) (a) Solve the differential equation, using \(T(0)=90\) and \(T(5)=70\), to find \(T\) in terms of \(t\) and the value of \(k\). [3]
2. (b) (b) Find the time taken for the tea to cool to 40\(^\circ\)C. [2]

A tank holds 200 litres of water in which 10 kg of salt is dissolved. Brine containing 0.5 kg of salt per litre flows in at 4 litres per minute. The mixture is kept well stirred and flows out at 4 litres per minute, so the volume stays at 200 litres. Let \(m\) kg be the mass of salt in the tank after \(t\) minutes. [6]

1. (a) (a) Show that \(\frac{dm}{dt}=2-\frac{m}{50}\). [2]
2. (b) (b) Solve the differential equation to find \(m\) in terms of \(t\). [2]
3. (c) (c) Find the time taken for the mass of salt to reach 60 kg. [1]
4. (d) (d) Write down the mass of salt in the tank in the long run. [1]

The slope field shows the differential equation \(\frac{dy}{dx}=\frac{x(y-2)}{2}\) for integer values of \(x\) and \(y\). [6]

1. (a) (a) Find the gradient of the solution curve at the points \((1,4)\) and \((-2,0)\). [2]
2. (b) (b) Show that \(y=2\) is a constant solution of the differential equation. [1]
3. (c) (c) Copy the slope field and sketch the solution curve that passes through \((0,1)\). [2]
4. (d) (d) Use your curve to estimate the value of \(y\) when \(x=2\). [1]

The population \(P\) (in thousands) of fish in a lake satisfies \(\frac{dP}{dt}=0.4P\left(1-\frac{P}{50}\right)\), where \(t\) is the time in years and \(P>0\). [6]

1. (a) (a) Find the two values of \(P\) for which \(\frac{dP}{dt}=0\). [2]
2. (b) (b) By finding the sign of \(\frac{dP}{dt}\) when \(P=40\) and when \(P=60\), decide which of the two values in part (a) is stable. Give a reason. [2]
3. (c) (c) Write down the value of \(P\) for which the population is growing fastest. [1]
4. (d) (d) Find the greatest rate of growth. [1]

The variables \(x\) and \(y\) measure how far the price and the stock level of a product are from their equilibrium values. They satisfy the system \(\frac{dx}{dt}=x - 5 y,\quad \frac{dy}{dt}=2 x - 3 y\). [6]

1. (a) (a) Find the eigenvalues of the matrix of the system. [2]
2. (b) (b) State the type of the equilibrium point at the origin, and describe what happens to \(x\) and \(y\) in the long run. [2]
3. (c) (c) Find \(\frac{dx}{dt}\) at the point \((0,1)\), and hence state whether the spiral is clockwise or anticlockwise. [2]

A pendulum swings so that its angle \(\theta\) radians from the vertical after \(t\) seconds satisfies \(\frac{d^{2}\theta}{dt^{2}}+5\sin\theta=0\), with \(\theta(0)=0.6\) and \(\theta'(0)=0\). [5]

1. (a) (a) Let \(\omega=\frac{d\theta}{dt}\). Write the equation as a pair of first order equations and use Euler's method with step length \(0.1\) to estimate \(\theta(0.2)\). Work in radians. [3]
2. (b) (b) A simpler model replaces \(\sin\theta\) by \(\theta\). Use Euler's method with the same step length to estimate \(\theta(0.2)\) for this model. [2]

A chemist models the rate of a reaction by \(\frac{dy}{dx}=xy^{2}\), where \(y\) is the amount of product in grams after \(x\) hours, and \(y(0)=1\). [14]

1. (a) (a) Use Euler's method with step length \(0.5\) to estimate \(y(1)\). [3]
2. (b) (b) Use Euler's method with step length \(0.25\) to estimate \(y(1)\). [3]
3. (c) (c) Solve the differential equation, giving \(y\) in terms of \(x\). [4]
4. (d) (d) Find the percentage error in your estimate from part (b). [2]
5. (e) (e) Explain why both estimates are smaller than the exact value, and say how to get a more accurate estimate. [2]

The numbers of aphids \(x\) and ladybirds \(y\) in a greenhouse, \(t\) weeks after they are first counted, are modelled by \(\frac{dx}{dt}=0.6x-0.01xy,\quad \frac{dy}{dt}=0.004xy-0.3y\), with \(x(0)=100\) and \(y(0)=40\). [14]

1. (a) (a) Write down the Euler recurrence relations for \(x_{n+1}\) and \(y_{n+1}\) with step length \(h\). [2]
2. (b) (b) Use Euler's method with \(h=0.5\) to estimate \(x(1)\) and \(y(1)\). [4]
3. (c) (c) Use technology and Euler's method with \(h=0.1\) to estimate \(x(4)\) and \(y(4)\). [3]
4. (d) (d) Use technology and Euler's method with \(h=0.1\) to find the largest value of \(y\) for \(0\leq t\leq10\). [3]
5. (e) (e) Euler's method with \(h=0.1\) gives successive peaks of \(x\) of \(133\) (at \(t=2.4\)) and \(143\) (at \(t=17.7\)). A student says that the model predicts that the peaks of \(x\) will keep growing. Comment on the student's statement. [2]

Two species of fish share a lake and help each other to grow. Their populations \(x\) and \(y\) in hundreds, after \(t\) years, satisfy \(\frac{dx}{dt}=3 x + 2 y,\quad \frac{dy}{dt}=x + 4 y\), with \(x(0)=40\) and \(y(0)=10\). [16]

1. (a) (a) Write the system in the form \(\frac{d}{dt}\begin{pmatrix}x\\y\end{pmatrix}=A\begin{pmatrix}x\\y\end{pmatrix}\), stating \(A\). [1]
2. (b) (b) Find the eigenvalues of \(A\). [2]
3. (c) (c) Find an eigenvector for each eigenvalue. [3]
4. (d) (d) Write down the general solution of the system. [2]
5. (e) (e) Find \(x(t)\) and \(y(t)\). [3]
6. (f) (f) State the type of the equilibrium point at the origin, and sketch the phase portrait, marking the directions of the eigenvectors. [3]
7. (g) (g) Find the ratio \(y:x\) as \(t\) becomes large. [2]

The displacement \(x\) cm of a car's suspension \(t\) seconds after a bump satisfies \(\frac{d^{2}x}{dt^{2}}+4\frac{dx}{dt}+3x=0\), with \(x(0)=2\) and \(x'(0)=-1\). [16]

1. (a) (a) Let \(y=\frac{dx}{dt}\). Write the equation as a pair of first order equations and give the matrix of the system. [2]
2. (b) (b) Use Euler's method with step length \(0.5\) to estimate \(x(1)\). [3]
3. (c) (c) Find the eigenvalues of the matrix, and hence decide whether the solution is stable. [3]
4. (d) (d) Find the exact solution \(x(t)\). [4]
5. (e) (e) Find the percentage error of your estimate from part (b). [2]
6. (f) (f) Decide whether the solutions of (i) \(\frac{d^{2}x}{dt^{2}}+2\frac{dx}{dt}-15x=0\) and (ii) \(\frac{d^{2}x}{dt^{2}}+2\frac{dx}{dt}+10x=0\) are stable. [2]

Two warehouses store a product. The stock levels \(x\) and \(y\) (in hundreds of items) \(t\) weeks after a delivery satisfy \(\frac{dx}{dt}=-3x+y+3,\ \frac{dy}{dt}=x-2y+4\). [8]

1. (a) (a) Find the coordinates of the equilibrium point. [2]
2. (b) (b) Find the eigenvalues of the matrix of coefficients, and hence classify the equilibrium point as stable or unstable. [3]
3. (c) (c) A manager claims that \((2,4)\) is an equilibrium point. Use one step of Euler's method with step length \(0.1\) from \((2,4)\) to test the claim. [2]
4. (d) (d) Write down the long-run values of \(x\) and \(y\) for any starting point. [1]