4.10b Model with differential equations: kinematics and other contexts

9 In 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 time \(t\) seconds after the start of the reaction are \(x , 10 - x\) and \(20 - x\) respectively. At any time the rate of formation of \(X\) is proportional to the product of the masses of \(Y\) and \(Z\) present at the time. When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.01 ( 10 - x ) ( 20 - x )$$
2. Solve this differential equation and obtain an expression for \(x\) in terms of \(t\).
3. State what happens to the value of \(x\) when \(t\) becomes large.

10 A certain substance is formed in a chemical reaction. The mass of substance formed \(t\) seconds after the start of the reaction is \(x\) grams. At any time the rate of formation of the substance is proportional to \(( 20 - x )\). When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.05 ( 20 - x ) .$$
2. Find, in any form, the solution of this differential equation.
3. Find \(x\) when \(t = 10\), giving your answer correct to 1 decimal place.
4. State what happens to the value of \(x\) as \(t\) becomes very large.

10 A large plantation of area \(20 \mathrm {~km} ^ { 2 }\) is becoming infected with a plant disease. At time \(t\) years the area infected is \(x \mathrm {~km} ^ { 2 }\) and the rate of increase of \(x\) is proportional to the ratio of the area infected to the area not yet infected. When \(t = 0 , x = 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 1\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 19 x } { 20 - x }$$
2. Solve the differential equation and show that when \(t = 1\) the value of \(x\) satisfies the equation \(x = \mathrm { e } ^ { 0.9 + 0.05 x }\).
3. Use an iterative formula based on the equation in part (b), with an initial value of 2 , to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
4. Calculate the value of \(t\) at which the entire plantation becomes infected.

1. In freezing temperatures, ice forms on the surface of the water in a barrel. At time \(t\) hours after the start of freezing, the thickness of the ice formed is \(x \mathrm {~mm}\). You may assume that the thickness of the ice is uniform across the surface of the water.

At 4 pm there is no ice on the surface, and freezing begins.
At 6pm, after two hours of freezing, the ice is 1.5 mm thick.
In a simple model, the rate of increase of \(x\), in mm per hour, is assumed to be constant for a period of 20 hours. Using this simple model,

1. express \(t\) in terms of \(x\),
2. find the value of \(t\) when \(x = 3\) In a second model, the rate of increase of \(x\), in mm per hour, is given by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \lambda } { ( 2 x + 1 ) } \text { where } \lambda \text { is a constant and } 0 \leqslant t \leqslant 20$$ Using this second model,
3. solve the differential equation and express \(t\) in terms of \(x\) and \(\lambda\),
4. find the exact value for \(\lambda\),
5. find at what time the ice is predicted to be 3 mm thick.

4. Given that $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y = 0$$

1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 28 } { y ^ { 2 } } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 3 } - \frac { 24 } { y } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right)$$ Given also that \(y = 8\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)
2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients where possible.

1. Given that

$$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 2 y = 0 \quad y > 0$$

1. determine \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(y\) Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) at \(x = 0\)
2. determine a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient in its simplest form.

3. A scientist is modelling the amount of a chemical in the human bloodstream. The amount \(x\) of the chemical, measured in \(\mathrm { mg } l ^ { - 1 }\), at time \(t\) hours satisfies the differential equation $$2 x \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 6 \left( \frac { \mathrm { dx } } { \mathrm { dt } } \right) ^ { 2 } = x ^ { 2 } - 3 x ^ { 4 } , \quad x > 0$$

1. Show that the substitution \(\mathrm { y } = \frac { 1 } { x ^ { 2 } }\) transforms this differential equation into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + y = 3$$
2. Find the general solution of differential equation \(I\). Given that at time \(t = 0 , x = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\),
3. find an expression for \(x\) in terms of \(t\),
4. write down the maximum value of \(x\) as \(t\) varies.

7 A quantity of oil is dropped into water. When the oil hits the water it spreads out as a circle. The radius of the circle is \(r \mathrm {~cm}\) after \(t\) seconds and when \(t = 3\) the radius of the circle is increasing at the rate of 0.5 centimetres per second.
One observer believes that the radius increases at a rate which is proportional to \(\frac { 1 } { ( t + 1 ) }\).

1. Write down a differential equation for this situation, using \(k\) as a constant of proportionality.
2. Show that \(k = 2\).
3. Calculate the radius of the circle after 10 seconds according to this model. Another observer believes that the rate of increase of the radius of the circle is proportional to \(\frac { 1 } { ( t + 1 ) ( t + 2 ) }\).
4. Write down a new differential equation for this new situation. Using the same initial conditions as before, find the value of the new constant of proportionality.
5. Hence solve the differential equation.
6. Calculate the radius of the circle after 10 seconds according to this model.

5 The variables \(x\) and \(y\) are related by the differential equation $$x ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x y + x + 1 .$$

1. Use the substitution \(y = u - \frac { 1 } { x }\), where \(u\) is a function of \(x\), to show that the differential equation may be written as $$x ^ { 2 } \frac { \mathrm {~d} u } { \mathrm {~d} x } = u .$$
2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y = \mathrm { f } ( x )\).

11 It is given that \(x \neq 0\) and $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 x y = 8 x ^ { 2 } + 16$$ Show that if \(z = x y\) then $$\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} x ^ { 2 } } + 4 z = 8 x ^ { 2 } + 16$$ Find \(y\) in terms of \(x\), given that \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2\) when \(x = \frac { 1 } { 2 } \pi\).

7 An engineer is modelling the motion of a particle \(P\) of mass 0.5 kg in a wind tunnel. \(P\) is modelled as travelling in a straight line. The point \(O\) is a fixed point within the wind tunnel. The displacement of \(P\) from \(O\) at time \(t\) seconds is \(x\) metres, for \(t \geqslant 0\). You are given that \(x \geqslant 0\) for all \(t \geqslant 0\) and that \(P\) does not reach the end of the wind tunnel.
If \(t \geqslant 0\), then \(P\) is subject to three forces which are modelled in the following way.

• The first force has a magnitude of \(5 ( t + 1 ) \cosh t \mathrm {~N}\) and acts in the positive \(x\)-direction.
• The second force has a magnitude of \(0.5 x \mathrm {~N}\) and acts towards \(O\).
• The third force has a magnitude of \(\left| \frac { d x } { d t } \right| \mathrm { N }\) and acts in the direction of motion of the particle.
  1. 1. Show that the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t\) is \(6 - 5 t\).
     2. Use your answer to part (a)(ii) to show that the term in \(t ^ { 2 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(- 3 t ^ { 2 }\).
     3. By differentiating the differential equation in part (a) with respect to \(t\), show that the term in \(t ^ { 3 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(0.5 t ^ { 3 }\). You are given that the complete Maclaurin series for the function f is valid for all values of \(t \geqslant 0\).
        After 0.25 seconds \(P\) has travelled 1.43 m towards the origin.
  2. 1. By using the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\), evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 0.25\).
     2. Explain why it might not be sensible to use the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\) to evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 10\).

8 A particle \(P\) of mass 2 kg can only move along the straight line segment \(O A\), where \(O A\) is on a rough horizontal surface. The particle is initially at rest at \(O\) and the distance \(O A\) is 0.9 m . When the time is \(t\) seconds the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). \(P\) is subject to a force of magnitude \(4 \mathrm { e } ^ { - 2 t } \mathrm {~N}\) in the direction of \(A\) for any \(t \geqslant 0\). The resistance to the motion of \(P\) is modelled as being proportional to \(v\). At the instant when \(t = \ln 2 , v = 0.5\) and the resultant force on \(P\) is 0 N .

1. Show that, according to the model, \(\frac { d v } { d t } + v = 2 e ^ { - 2 t }\).
2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\).
3. By considering the behaviour of \(v\) as \(t\) becomes large explain why, according to the model, \(P\) 's speed must reach a maximum value for some \(t > 0\).
4. Determine the maximum speed considered in part (c).
5. Determine the greatest value of \(t\) for which the model is valid.

4 The resistive force, \(F\), on a sphere falling through a viscous fluid is thought to depend on the radius of the sphere, \(r\), the velocity of the sphere, \(v\), and the viscosity of the fluid, \(\eta\). You are given that \(\eta\) is measured in \(\mathrm { N } \mathrm { m } ^ { - 2 } \mathrm {~s}\).

1. By considering its units, find the dimensions of viscosity. A model of the resistive force suggests the following relationship: \(\mathbf { F } = 6 \pi \eta ^ { \alpha } \mathbf { r } ^ { \beta } \mathbf { v } ^ { \gamma }\).
2. Explain whether or not it is possible to use dimensional analysis to verify that the constant \(6 \pi\) is correct.
3. Use dimensional analysis to find the values of \(\alpha , \beta\) and \(\gamma\). A sphere of radius \(r\) and mass \(m\) falls vertically from rest through the fluid. After a time \(t\) its velocity is \(v\).
4. By setting up and solving a differential equation, show that \(\mathrm { e } ^ { - \mathrm { kt } } = \frac { \mathrm { g } - \mathrm { kv } } { \mathrm { g } }\) where \(\mathrm { k } = \frac { 6 \pi \eta \mathrm { r } } { \mathrm { m } }\). As the time increases, the velocity of the sphere tends towards a limit called the terminal velocity.
5. Find, in terms of \(g\) and \(k\), the terminal velocity of the sphere. In a sequence of experiments the sphere is allowed to fall through fluids of different viscosity, ranging from small to very large, with all other conditions being constant. The terminal velocity of the sphere through each fluid is measured.
6. Describe how, according to the model, the terminal velocity of the sphere changes as the viscosity of the fluid through which it falls increases.

3 A particle is oscillating in a vertical line. At time \(t\) seconds, its displacement above the centre of the oscillations is \(x\) metres, where \(x = A \sin \omega t + B \cos \omega t\) (and \(A , B\) and \(\omega\) are constants).

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \omega ^ { 2 } x\). When \(t = 0\), the particle is 2 m above the centre of the oscillations, the velocity is \(1.44 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards, and the acceleration is \(0.18 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards.
2. Find \(A , B\) and \(\omega\).
3. Show that the period of oscillation is 20.9 s (correct to 3 significant figures), and find the amplitude.
4. Find the total distance travelled by the particle between \(t = 12\) and \(t = 24\).

3 An aeroplane is taking off from a runway. It starts from rest. The resultant force in the direction of motion has power, \(P\) watts, modelled by $$P = 0.0004 m \left( 10000 v + v ^ { 3 } \right) ,$$ where \(m \mathrm {~kg}\) is the mass of the aeroplane and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity at time \(t\) seconds. The displacement of the aeroplane from its starting point is \(x \mathrm {~m}\). To take off successfully the aeroplane must reach a speed of \(80 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before it has travelled 900 m .

1. Formulate and solve a differential equation for \(v\) in terms of \(x\). Hence show that the aeroplane takes off successfully.
2. Formulate a differential equation for \(v\) in terms of \(t\). Solve the differential equation to show that \(v = 100 \tan ( 0.04 t )\). What feature of this result casts doubt on the validity of the model?
3. In fact the model is only valid for \(0 \leqslant t \leqslant 11\), after which the power remains constant at the value attained at \(t = 11\). Will the aeroplane take off successfully?

5. A rocket is launched vertically upwards under gravity from rest at time \(t = 0\). The rocket propels itself upward by ejecting burnt fuel vertically downwards at a constant speed \(u\) relative to the rocket. The initial mass of the rocket, including fuel, is \(M\). At time \(t\), before all the fuel has been used up, the mass of the rocket, including fuel, is \(M ( 1 - k t )\) and the speed of the rocket is \(v\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { k u } { 1 - k t } - g\).
2. Hence find the speed of the rocket when \(t = \frac { 1 } { 3 k }\).

6. A rocket-driven car moves along a straight horizontal road. The car has total initial mass \(M\). It propels itself forwards by ejecting mass backwards at a constant rate \(\lambda\) per unit time at a constant speed \(U\) relative to the car. The car starts from rest at time \(t = 0\). At time \(t\) the speed of the car is \(v\). The total resistance to motion is modelled as having magnitude \(k v\), where \(k\) is a constant. Given that \(t < \frac { M } { \lambda }\), show that

1. \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }\),
2. \(v = \frac { \lambda U } { k } \left\{ 1 - \left( 1 - \frac { \lambda t } { M } \right) ^ { \frac { k } { \lambda } } \right\}\).
   (6)
   (Total 13 marks)

7. A motor boat of mass \(M\) is moving in a straight line, with its engine switched off, across a stretch of still water. The boat is moving with speed \(U\) when, at time \(t = 0\), it develops a leak. The water comes in at a constant rate so that at time \(t\), the mass of water in the boat is \(\lambda t\). At time \(t\) the speed of the boat is \(v\) and it experiences a total resistance to motion of magnitude \(2 \lambda v\).

1. Show that \(( M + \lambda t ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 \lambda v = 0\).
   (6)
2. Show that the time taken for the speed of the boat to reduce to \(\frac { 1 } { 2 } U\) is \(\frac { M } { \lambda } \left( 2 ^ { \frac { 1 } { 3 } } - 1 \right)\).
   (6) The boat sinks when the mass of water inside the boat is \(M\).
3. Show that the boat does not sink before the speed of the boat is \(\frac { 1 } { 2 } U\).

1. A particle moves in a plane in such a way that its position vector \(\mathbf { r }\) metres at time \(t\) seconds satisfies the differential equation

$$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } = \mathbf { 0 }$$ When \(t = 0\), the particle is at the origin and is moving with velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find \(\mathbf { r }\) in terms of \(t\).

1. Solve the differential equation

$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } - 2 \mathbf { r } = \mathbf { 0 }$$ given that when \(t = 0 , \mathbf { r } . \mathbf { j } = 0\) and \(\mathbf { r } \times \mathbf { j } = \mathbf { i } + \mathbf { k }\).

17 A cyclist accelerates from rest for 5 seconds then brakes for 5 seconds, coming to rest at the end of the 10 seconds. The total mass of the cycle and rider is \(m \mathrm {~kg}\), and at time \(t\) seconds, for \(0 \leqslant t \leqslant 10\), the cyclist's velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance to motion, modelled by a force of magnitude 0.1 mvN , acts on the cyclist during the whole 10 seconds.

1. Explain why modelling the resistance to motion in this way is likely to be more realistic than assuming this force is constant. During the braking phase of the motion, for \(5 \leqslant t \leqslant 10\), the brakes apply an additional constant resistance force of magnitude \(2 m \mathrm {~N}\) and the cyclist does not provide any driving force.
2. Show that, for \(5 \leqslant t \leqslant 10 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = - 2\).
3. 1. Solve the differential equation in part (b).
   2. Hence find the velocity of the cyclist when \(t = 5\). During the acceleration phase ( \(0 \leqslant t \leqslant 5\) ), the cyclist applies a driving force of magnitude directly proportional to \(t\).
4. Show that, for \(0 \leqslant t \leqslant 5 , \frac { \mathrm {~d} v } { \mathrm {~d} t } + 0.1 v = \lambda t\), where \(\lambda\) is a positive constant.
5. 1. Show by integration that, for \(0 \leqslant t \leqslant 5 , v = 10 \lambda \left( t - 10 + 10 \mathrm { e } ^ { - 0.1 t } \right)\).
   2. Hence find \(\lambda\).
6. Find the total distance, to the nearest metre, travelled by the cyclist during the motion.

17 In a chemical process, a vessel contains 1 litre of pure water. A liquid chemical is then passed into the top of the vessel at a constant rate of \(a\) litres per minute and thoroughly mixed with the water. At the same time, the resulting mixture is drawn from the bottom of the vessel at a constant rate of \(b\) litres per minute. You may assume that the chemical mixes instantly and uniformly with the water. After \(t\) minutes, the mixture in the vessel contains \(x\) litres of the chemical.

1. 1. Show that the proportion of chemical present in the vessel after \(t\) minutes is $$\frac { x } { 1 + ( a - b ) t } .$$
   2. Hence show that \(\frac { d x } { d t } + \frac { b x } { 1 + ( a - b ) t } = a\).
2. First, consider the case where \(\mathbf { b } = \mathbf { a }\).
   1. Solve the differential equation to find \(x\) in terms of \(a\) and \(t\).
   2. Given that after 1 minute the vessel contains equal amounts of water and chemical, find the rate of inflow of chemical.
3. Now consider the case where \(\mathrm { b } = 2 \mathrm { a }\).
   1. Explain why the differential equation in part (a)(ii) is now invalid for \(\mathrm { t } \geqslant \frac { 1 } { \mathrm { a } }\).
   2. Find the maximum amount of chemical in the vessel.

3 This question concerns the family of differential equations \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - x ^ { a } y \left( { } ^ { * } \right)\) where \(a\) is \(- 1,0\) or 1 .

1. Determine and describe geometrically the isoclines of (\textit{) when
   1. \(a = - 1\),
   2. \(a = 0\),
   3. \(a = 1\).
2. In this part of the question \(a = 0\).
   1. Write down the solution to \(( * )\) which passes through the point \(( 0 , b )\) where \(b \neq 1\).
   2. Write down the equation of the asymptote to this solution.
3. In this part of the question \(a = - 1\).
   1. Write down the solution to \(( * )\) which passes through the point \(( c , d )\) where \(c \neq 0\).
   2. Describe the relationship between \(c\) and \(d\) when the solution in part (i) has a stationary point.
4. In this part of the question \(a = 1\).
   1. The standard Runge-Kutta method of order 4 for the solution of the differential equation \(\mathrm { f } ( x , y ) = \frac { \mathrm { d } y } { \mathrm {~d} x }\) is as follows. \(k _ { 1 } = h \mathrm { f } \left( x _ { n } , y _ { n } \right)\) \(k _ { 2 } = h \mathrm { f } \left( x _ { n } + \frac { h } { 2 } , y _ { n } + \frac { k _ { 1 } } { 2 } \right)\) \(k _ { 3 } = h \mathrm { f } \left( x _ { n } + \frac { h } { 2 } , y _ { n } + \frac { k _ { 2 } } { 2 } \right)\) \(k _ { 4 } = h \mathrm { f } \left( x _ { n } + h , y _ { n } + k _ { 3 } \right)\) \(y _ { n + 1 } = y _ { n } + \frac { 1 } { 6 } \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right)\).
      Construct a spreadsheet to solve (}) in the case \(x _ { 0 } = 0\) and \(y _ { 0 } = 1.5\). State the formulae you have used in your spreadsheet.
   2. Use your spreadsheet with \(h = 0.05\) to find an approximation to the value of \(y\) when \(x = 1\).
   3. The solution to \(( * )\) in which \(x _ { 0 } = 0\) and \(y _ { 0 } = 1.5\) has a maximum point ( \(r , s\) ) with \(0 < r < 1\). Use your spreadsheet with suitable values of \(h\) to estimate \(r\) to two decimal places. Justify your answer.

3 In this question you are required to consider the family of differential equations \(\frac { d y } { d x } = \frac { y ^ { a } } { x + 1 } - \frac { 1 } { y } ( * )\) and its solutions. The parameter \(a\) is a real number. You should assume that \(x \geqslant 0\) and \(y > 0\) throughout this question.

1. In this part of the question \(a = 1\).
   1. On the axes in the Printed Answer Booklet
      • Sketch the isocline defined by \(\frac { d y } { d x } = 0\).
      • Shade and label the region in which \(\frac { \mathrm { dy } } { \mathrm { dx } } > 0\).
      • Shade and label the region in which \(\frac { \mathrm { dy } } { \mathrm { dx } } < 0\).
      • For \(b > 0\), find, in terms of \(b\), the solution to \(( * )\) which passes through the point \(( 0 , b )\).
      • Determine
      • The values of \(b > 0\) for which the solution in (ii) has a turning point.
      • The corresponding maximum value of \(y\).
      • Fig. 3.1 and Fig. 3.2 show tangent fields for two distinct but unspecified values of \(a\). In each case a sketch of the solution curve \(y = \mathrm { g } ( x )\) which passes through \(( 0,2 )\) is shown for \(0 \leqslant x \leqslant 0.5\).
      \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{43fdb360-0f80-4794-917c-f28b04181fa4-4_656_648_1777_301} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{43fdb360-0f80-4794-917c-f28b04181fa4-4_656_652_1777_1117} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
      \end{figure}
      1. For the case in Fig. 3.1 suggest a possible value of \(a\).
      2. For the case in Fig. 3.2 suggest a possible value of \(a\).
      3. In each case, continue the sketch of the solution curves for \(0.5 \leqslant x \leqslant 5\) in the Printed Answer Booklet.
      4. State a feature which is present in one of the curves in part (iii) for \(0.5 \leqslant x \leqslant 5\) but not in the other.
2. 1. The Euler method for the solution of the differential equation \(\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { f } ( x , y )\) is as follows $$y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } \right)$$ It is given that \(x _ { 0 } = 0\) and \(y _ { 0 } = 2\).
      Use your spreadsheet with \(h = 0.1\) to approximate the value of \(y\) when \(x = 3\) for the solution to (*) in which \(y = 2\) when \(x = 0\).
   2. In this part of the question \(a = - 0.2\). Use your spreadsheet to approximate, to \(\mathbf { 1 }\) decimal place, the \(x\)-coordinate of the local maximum for the solution to (*) in which \(y = 2\) when \(x = 0\).

4 In this question you are required to consider the family of differential equations $$\frac { d P } { d t } = r P \left( 1 - \frac { P } { K } \right) , \quad t \geqslant 0 , \quad P ( t ) \geqslant 0 \left( ^ { * } \right)$$ where \(r\) and \(K\) are positive constants. This differential equation can be used as a model for the size of a population \(P\) as a function of time \(t\).

1. 1. Determine the values of \(P\) for which
      $$\frac { d P } { d t } = 2 P ^ { 1.25 } \left( 1 - \frac { P } { 1000 } \right) ^ { 1.5 } , t \geqslant 0 , P ( t ) \geqslant 0 ( * * )$$ The diagram shows the tangent field to (**), and a solution in which \(P = 1\) when \(t = 0\), produced using a much more accurate numerical method. \includegraphics[max width=\textwidth, alt={}, center]{4715d0f0-a860-4189-802f-1d2d019e1115-4_899_1552_1763_319}
      1. The Euler method for the solution of the differential equation \(f ( t , P ) = \frac { d P } { d t }\) is as follows $$P _ { n + 1 } = P _ { n } + h f \left( t _ { n } , P _ { n } \right)$$ It is given that \(t _ { 0 } = 0\) and \(P _ { 0 } = 1\).