Differential equations – further maths worksheet with answers to download as PDF

Solve each differential equation by finding an integrating factor \(I=e^{\int P\,dx}\), where the equation is written as \(\dfrac{dy}{dx}+Py=Q\).

1. \(\dfrac{dy}{dx}+3y=6\)
2. \(x\dfrac{dy}{dx}+2y=4x\) for \(x>0\)
3. \(\dfrac{dy}{dx}+y\tan x=\cos x\) for \(-\tfrac\pi2\lt x\lt \tfrac\pi2\)
4. \(\dfrac{dy}{dx}+2y=e^{-x}\), given that \(y=3\) when \(x=0\)

Find the general solution of each equation by writing down and solving the auxiliary equation.

1. \(\dfrac{d^2y}{dx^2}-5\dfrac{dy}{dx}+6y=0\)
2. \(\dfrac{d^2y}{dx^2}+6\dfrac{dy}{dx}+9y=0\)
3. \(\dfrac{d^2y}{dx^2}+4\dfrac{dy}{dx}+13y=0\)
4. \(\dfrac{d^2y}{dx^2}+16y=0\)

In each part find a particular integral, then write down the general solution.

1. \(y''-4y'+3y=6x\)
2. \(y''+y'-6y=10e^{2x}\) (check the roots of the auxiliary equation first)
3. \(y''+9y=\sin2x\)

Solve \(\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}-2y=0\) given that \(y=1\) and \(\dfrac{dy}{dx}=5\) when \(x=0\).

1. Write down the auxiliary equation and solve it.
2. Write down the general solution.
3. Use the two conditions to find \(A\) and \(B\).
4. State the value of \(y\) when \(x=\ln2\).

A particle moves so that \(\dfrac{d^2x}{dt^2}=-16x\), with \(x=3\) and \(\dfrac{dx}{dt}=0\) when \(t=0\) (metres and seconds).

1. Write down the general solution and find the particular solution.
2. State the amplitude and the period of the motion.
3. Find the greatest speed of the particle.
4. Find the speed of the particle when \(x=2\).

Consider the equation \(y''+4y'+4y=0\).

1. Show that \(y=e^{-2x}\) satisfies the equation.
2. Show that \(y=xe^{-2x}\) satisfies the equation.
3. Write down the general solution and explain why both functions are needed.
4. Find the solution with \(y(0)=1\) and \(y'(0)=0\).

A cup of tea at \(90^\circ\)C is left in a room at a constant \(20^\circ\)C. Newton’s law of cooling says that the temperature \(T\) degrees Celsius after \(t\) minutes satisfies \(\dfrac{dT}{dt}=-k(T-20)\) for a constant \(k>0\). After 5 minutes the tea is at \(60^\circ\)C.

1. Solve the differential equation with \(T(0)=90\).
2. Show that \(k=\tfrac15\ln\tfrac74\approx0.112\).
3. How long after being made does the tea reach \(40^\circ\)C, the temperature at which Jo likes to drink it?
4. What happens to \(T\) as \(t\to\infty\)?

A drug is fed into a patient’s blood by a drip at a constant rate. The concentration \(C\) mg per litre at time \(t\) hours satisfies \(\dfrac{dC}{dt}=4-0.2C\), with \(C=0\) at \(t=0\).

1. Write the equation in the form \(\dfrac{dC}{dt}+PC=Q\) and find the integrating factor.
2. Solve the equation to show that \(C=20\left(1-e^{-0.2t}\right)\).
3. Find the limiting concentration as \(t\to\infty\).
4. The doctor wants a concentration of 15 mg per litre. After how many hours does it reach this value?

A tank holds 200 litres of water in which 10 kg of salt is dissolved. Brine containing 0.1 kg of salt per litre flows in at 4 litres per minute. The well-stirred mixture flows out at 4 litres per minute, so the volume stays at 200 litres.

1. Show that the mass \(Q\) kg of salt in the tank after \(t\) minutes satisfies \(\dfrac{dQ}{dt}=0.4-\dfrac Q{50}\).
2. Solve the equation to show that \(Q=20-10e^{-t/50}\).
3. How long does it take for the tank to hold 15 kg of salt?
4. Find the limit of \(Q\) and explain it in terms of the concentration.

In a circuit a capacitor of \(500\ \mu\)F is charged through a resistor of \(2000\ \Omega\) from a 12 V battery. The voltage \(V\) volts across the capacitor satisfies \(RC\dfrac{dV}{dt}+V=12\), with \(V=0\) when \(t=0\) seconds.

1. Calculate \(RC\) and write the equation in standard form.
2. Solve for \(V\) in terms of \(t\).
3. How long does the capacitor take to reach 9 V?
4. Find \(\dfrac{dV}{dt}\) at \(t=0\) and describe what it tells you about the start of charging.

The suspension of a scooter is modelled by a mass of 2 kg attached to a spring and a damper. The displacement \(x\) metres from equilibrium after \(t\) seconds satisfies \(2\dfrac{d^2x}{dt^2}+8\dfrac{dx}{dt}+26x=0\). The scooter is released at \(x=0.5\) from rest.

1. Show that the auxiliary equation has roots \(-2\pm3i\) and say whether the damping is light, critical or heavy.
2. Find \(x\) in terms of \(t\).
3. State the period of the oscillations and the factor by which the amplitude is reduced after each period.
4. Explain what would change if the damper were replaced by a stronger one with \(c=20\) in place of \(8\).

Ben has £40 000 in an account that earns 4% interest per year, added continuously, and he withdraws £2000 per year at a steady rate. The balance £\(A\) after \(t\) years satisfies \(\dfrac{dA}{dt}=0.04A-2000\).

1. Find the equilibrium balance, at which the balance would neither grow nor fall.
2. Solve the equation with \(A(0)=40000\).
3. After how many years is the account empty?
4. What initial balance would keep the account steady for ever?

A footbridge deck behaves like a mass on a spring and its vertical displacement \(x\) metres satisfies \(\dfrac{d^2x}{dt^2}+4x=F(t)\), where \(F\) is the forcing due to the footsteps of walkers.

1. Find the complementary function.
2. The walkers produce \(F=10\cos3t\). Find a particular integral of the form \(a\cos3t\) and write down the general solution.
3. If the walkers instead produce \(F=8\cos2t\), explain why the form \(a\cos2t+b\sin2t\) fails, and show that \(x=2t\sin2t\) is a particular integral.
4. What does the second answer say about the displacement as time passes, and why do soldiers break step when they cross a bridge?

A skydiver of mass 70 kg falls from rest. The air resistance is proportional to speed and the speed \(v\) m s\(^{-1}\) after \(t\) seconds satisfies \(\dfrac{dv}{dt}=9.8-0.2v\).

1. Find the terminal speed.
2. Solve the equation to show that \(v=49\left(1-e^{-0.2t}\right)\).
3. Find the speed after 10 seconds.
4. Integrate to find the distance fallen after 10 seconds.

A curve satisfies the differential equation \(x\dfrac{dy}{dx}-2y=x^4\) for \(x>0\).

1. Write the equation in the form \(\dfrac{dy}{dx}+Py=Q\) and find the integrating factor.
2. Find the general solution.
3. The curve passes through \((1,3)\). Find its equation.
4. Find the value of \(y\) at \(x=2\) and show that the curve has no stationary point for \(x>0\).

Consider \(\dfrac{d^2y}{dx^2}+2\dfrac{dy}{dx}+5y=5x+12\).

1. Find the complementary function.
2. Find a particular integral of the form \(ax+b\).
3. Write down the general solution.
4. Find the solution with \(y=3\) and \(\dfrac{dy}{dx}=0\) at \(x=0\), and describe the behaviour of \(y\) for large \(x\).

A mass of 1 kg moves on a spring of stiffness 25 N m\(^{-1}\) and is damped by a force \(c\dot x\) newtons. Its displacement \(x\) metres satisfies \(\ddot x+c\dot x+25x=0\).

1. Find the value of \(c\) for which the damping is critical.
2. For \(c=6\) find the roots of the auxiliary equation and say whether the motion oscillates.
3. For \(c=12\) show that there are two distinct real roots and write the general solution.
4. For \(c=10\), with \(x(0)=0\) and \(\dot x(0)=2\), solve the equation, and find the greatest displacement.

Consider the differential equation \(\dfrac{dy}{dx}+y=xe^{-x}\), with \(y=1\) when \(x=0\).

1. Find the integrating factor and show that \(\dfrac d{dx}(ye^x)=x\).
2. Find \(y\) in terms of \(x\).
3. Show that \(\dfrac{dy}{dx}=-\tfrac12(x-1)^2e^{-x}\).
4. Hence describe the shape of the curve for \(x\ge0\), including the nature of the point where \(x=1\).

Two quantities \(x\) and \(y\) vary with time \(t\) according to \(\dfrac{dx}{dt}=-x+2y\) and \(\dfrac{dy}{dt}=-2x-y\), with \(x=1\) and \(y=0\) at \(t=0\).

1. Show that \(y=\tfrac12\left(\dfrac{dx}{dt}+x\right)\) and use it to show that \(\dfrac{d^2x}{dt^2}+2\dfrac{dx}{dt}+5x=0\).
2. Solve for \(x\) in terms of \(t\).
3. Find \(y\) in terms of \(t\).
4. Describe how the point \((x,y)\) moves in the plane as \(t\) increases.

A particle of unit mass satisfies \(\ddot x+2\lambda\dot x+\omega^2x=0\) with \(\lambda\ge0\). Its energy is \(E=\tfrac12\dot x^2+\tfrac12\omega^2x^2\).

1. Show that \(\dfrac{dE}{dt}=-2\lambda\dot x^2\).
2. Deduce that \(E\) is constant when \(\lambda=0\) and decreasing when \(\lambda>0\).
3. For \(\lambda=0\) and \(x=3\cos4t\), verify that \(E\) is constant and find its value.
4. For the scooter in Question 11, \(x=e^{-2t}\left(\tfrac12\cos3t+\tfrac13\sin3t\right)\) with \(\lambda=2\) and \(\omega^2=13\). Verify numerically that \(\dfrac{dE}{dt}=-4\dot x^2\) at \(t=1\).