Differential equations – university Year 1 maths worksheet with answers to download as PDF

Solve each differential equation by separating the variables. Give \(y\) explicitly where you can, and use the initial condition when one is given.

1. \(\dfrac{dy}{dx}=3x^2y\) [3 marks]
2. \(\dfrac{dy}{dx}=e^{x-y}\) [3 marks]
3. \(x\dfrac{dy}{dx}=y\), with \(y(1)=3\) [3 marks]
4. \((1+x^2)\dfrac{dy}{dx}=xy\), with \(y(0)=2\) [4 marks]

Solve each first-order linear equation using an integrating factor.

1. \(y'+3y=6\) [3 marks]
2. \(x\dfrac{dy}{dx}+2y=x^2\) for \(x>0\) [4 marks]
3. \(y'+y=e^{-x}\), with \(y(0)=2\) [4 marks]

Find the general solution of each equation. State whether the roots of the auxiliary equation are real and distinct, repeated, or complex.

1. \(y''-7y'+12y=0\) [3 marks]
2. \(y''+6y'+9y=0\) [3 marks]
3. \(y''+2y'+10y=0\) [3 marks]

Solve each initial value problem.

1. \(y''-4y=0\), with \(y(0)=0\) and \(y'(0)=4\) [4 marks]
2. \(y''+4y'+13y=0\), with \(y(0)=2\) and \(y'(0)=-1\) [5 marks]

Find a particular integral for each equation, and hence write down the general solution of (a).

1. \(y''-y'-6y=12\) [3 marks]
2. \(y''+9y=18\cos2x\) [3 marks]
3. \(y''-3y'+2y=e^{x}\) [4 marks]

(a) State the order of each equation and whether it is linear: (i) \(y'''+xy=0\), (ii) \(y'=y^2\), (iii) \(y''+\sin y=0\). (b) Show that \(y=xe^{2x}\) solves \(y''-4y'+4y=0\). (c) Find \(A\) and \(B\) so that \(y=Ae^x+Be^{-x}\) satisfies \(y(0)=3\) and \(y'(0)=1\). [8 marks in total]

A culture in a university laboratory contains 500 bacteria. Their number \(P\) grows at a rate proportional to \(P\), so \(\dfrac{dP}{dt}=rP\), where \(t\) is the time in hours. After 3 hours there are 1 500 bacteria.

1. Solve the equation to show that \(P=500e^{rt}\), and find \(r\) to three significant figures.
2. Find the doubling time.
3. How many bacteria are there after 8 hours?
4. The lab must stop the experiment when there are 1 000 000 bacteria. After how many hours is that?

Newton's law of cooling says that the temperature \(T\) (in \(^\circ\)C) of an object in a room at temperature \(T_s\) satisfies \(\dfrac{dT}{dt}=-k(T-T_s)\). A cup of coffee at \(85^\circ\)C is left in a room at \(21^\circ\)C. After 5 minutes it has cooled to \(68^\circ\)C.

1. Show that \(T=21+64e^{-kt}\).
2. Find \(k\).
3. Find the temperature after 10 minutes.
4. It is safe to drink at \(40^\circ\)C. When is that?

A tank holds 200 litres of pure water. Brine containing 0.4 kg of salt per litre flows in at 5 litres per minute. The well-stirred mixture flows out at 5 litres per minute. Let \(Q\) be the mass of salt in kg after \(t\) minutes.

1. Explain why \(\dfrac{dQ}{dt}=2-\dfrac Q{40}\).
2. Solve the equation with \(Q(0)=0\).
3. How long does it take for the tank to hold 50 kg of salt?
4. What happens to \(Q\) in the long run?

In an electrical circuit a battery of voltage \(E=12\) V charges a capacitor through a resistor. The charge \(Q\) (in coulombs) satisfies \(R\dfrac{dQ}{dt}+\dfrac QC=E\). Here \(R=4000\ \Omega\) and \(C=250\ \mu\)F \(=2.5\times10^{-4}\) F, so \(RC=1\) second. At \(t=0\) the capacitor is empty.

1. Show that \(Q=CE\left(1-e^{-t}\right)\).
2. Write down the voltage across the capacitor, \(V=Q/C\), and the current \(I=\dfrac{dQ}{dt}\).
3. How long does it take for the voltage to reach 9 V?
4. What is the current at \(t=0\)?

A block of mass 0.5 kg hangs on a spring of stiffness \(k=32\) N/m. It is pulled 0.1 m below its rest position and released from rest. The displacement \(x\) metres satisfies \(0.5x''=-32x\) if there is no resistance.

1. Show that \(x''+64x=0\), and solve it.
2. Find the period and the maximum speed.
3. A damper is now added, giving \(0.5x''+2x'+32x=0\). Find \(m\) in the auxiliary equation and describe the motion.
4. By what factor does the amplitude shrink each second in the damped motion?

A drug is given through a drip at a constant rate of 4 mg per hour. The body removes the drug at a rate proportional to the amount \(C\) (in mg) in the blood, with constant 0.2 per hour. So \(\dfrac{dC}{dt}=4-0.2C\), and \(C(0)=0\).

1. Solve the equation.
2. Find the steady level that the amount approaches.
3. When does the amount reach 15 mg?
4. The doctor wants 15 mg after only 2 hours by giving an initial dose \(C(0)=C_0\) with the same drip. Find \(C_0\) to 2 decimal places.

An account earns interest continuously at 4% per year, and its owner also pays in £2 400 per year, spread evenly through the year. The balance \(B\) (in £) after \(t\) years satisfies \(\dfrac{dB}{dt}=0.04B+2400\). The opening balance is £1 000.

1. Solve the equation for \(B\).
2. Find the balance after 10 years.
3. How long until the balance reaches £100 000?

A skydiver of mass 70 kg falls with air resistance proportional to speed. Taking \(g=9.8\) m/s\(^2\) and a drag constant of 14 kg/s, Newton's second law gives \(70\dfrac{dv}{dt}=70\times9.8-14v\), so \(\dfrac{dv}{dt}=9.8-0.2v\). She starts from rest.

1. Solve for \(v(t)\).
2. Find the terminal velocity.
3. When does she reach 40 m/s?
4. Find the distance fallen in the first 10 seconds.

The number of fish \(P\) in a lake grows according to the logistic equation \(\dfrac{dP}{dt}=0.3P\left(1-\dfrac P{2000}\right)\), where \(t\) is in years. At \(t=0\) there are 200 fish.

1. Use the substitution \(u=\dfrac1P\) or partial fractions to show that \(P=\dfrac{2000}{1+9e^{-0.3t}}\).
2. When does the population reach 1 000?
3. The lake is then fished at a constant rate of 100 fish per year, so \(\dfrac{dP}{dt}=0.3P\left(1-\dfrac P{2000}\right)-100\). Find the two equilibrium populations.
4. Which equilibrium is stable? What happens if the population falls below the other one?

A quarter of a car of mass \(m=400\) kg is supported by a spring of stiffness \(k=40\,000\) N/m and a damper with damping constant \(c\) (N s/m). The vertical displacement \(x\) metres from equilibrium satisfies \(400x''+cx'+40\,000x=0\).

1. Show that the undamped angular frequency is 10 rad/s.
2. Find the critical value of \(c\), that is the value for which the auxiliary equation has a repeated root.
3. For \(c=2000\) show that the roots are complex, and write the general solution.
4. For \(c=12\,000\) find the roots. Explain why the car does not oscillate.
5. Which type of damping gives the fastest return to equilibrium without bouncing? Explain.

Engineers model the sideways sway \(x\) (in cm) of a footbridge by \(x''+4x=3\cos\omega t\), where the forcing comes from the footsteps of walkers. The bridge starts at rest, so \(x(0)=0\) and \(x'(0)=0\).

1. Find the complementary function.
2. For \(\omega=1\) find a particular integral and then the solution satisfying the conditions.
3. For \(\omega=2\) explain why the usual guess fails, find a particular integral of the form \(kt\sin2t\), and solve the problem.
4. Describe what happens to the sway in part (c) as time passes. Why do engineers avoid this situation?

There are 1 000 students on a campus. Let \(I\) be the number who have heard a rumour after \(t\) days. Each student who knows the rumour tells others at a rate proportional to the number who do not know it, so \(\dfrac{dI}{dt}=0.0004\,I(1000-I)\). Five students start the rumour: \(I(0)=5\).

1. Show that \(\dfrac{dI}{dt}=0.4I\left(1-\dfrac I{1000}\right)\).
2. Solve the equation and show that \(I=\dfrac{1000}{1+199e^{-0.4t}}\).
3. When have 500 students heard the rumour?
4. At what rate is the rumour spreading at that moment? Explain why this is the fastest rate.

A cylindrical tank has cross-sectional area \(2\) m\(^2\). Water leaves through a small hole of area \(0.002\) m\(^2\) at the bottom. Torricelli's law gives the outflow speed as \(\sqrt{2gh}\), so the height \(h\) (in m) satisfies \(2\dfrac{dh}{dt}=-0.002\sqrt{19.6\,h}\) (using \(g=9.8\)). Initially \(h=4\).

1. Show that \(\dfrac{dh}{dt}=-0.001\sqrt{19.6}\sqrt h\).
2. Solve the equation to find \(\sqrt h\) as a function of \(t\).
3. How long, to the nearest minute, does the tank take to empty?
4. How long does it take to drop from 4 m to 1 m?

Consider \(\dfrac{dy}{dx}=\dfrac yx+\dfrac xy\) for \(x>0\), \(y>0\), with \(y(1)=2\).

1. Put \(y=vx\) and show that \(x\dfrac{dv}{dx}=\dfrac1v\).
2. Solve for \(v\), and hence find \(y\) in terms of \(x\).
3. Find \(y(e)\) exactly.
4. Check your answer by differentiating.