Question 1

Marks: 5

Consider the first-order differential equation

$\frac{d y}{d x} - 5 x^{4} = 3$

Solve the equation given that $y = 40$ when $x = 2$ , giving your answer in the form $y = f (x)$ .

Key Concepts

Integrating Powers of x
Finding the Constant of Integration

Question 2a

Marks: 4

Use separation of variables to solve each of the following differential equations for $y$ :

a)

\frac{d y}{d x} = \frac{4 x^{2}}{y^{4}}

Key Concepts

Separation of Variables

Question 2b

Marks: 1

b)

\frac{d y}{d x} = (x^{2} + 1) e^{- y}

Key Concepts

Separation of Variables

Question 3a

Marks: 1

Use separation of variables to solve each of the following differential equations for which satisfies the given boundary condition:

a)

\frac{d y}{d x} = x y^{2}; y (2) = 1

Key Concepts

Separation of Variables
Finding the Constant of Integration

Question 3b

Marks: 5

b)

(x + 3) \frac{d y}{d x} = \sec y; y (- 2) = \frac{3 π}{2}

Key Concepts

Separation of Variables
Finding the Constant of Integration

Question 4a

Marks: 1

At any point in time, the rate of growth of a colony of bacteria is proportional to the current population size. At time $t = 0$ hours, the population size is 5000.

a)

Write a differential equation to model the size of the population of bacteria.

Key Concepts

Modelling with Differential Equations
Separation of Variables

Question 4b

Marks: 6

After 1 hour, the population has grown to 7000.

b)

By first solving the differential equation from part (a), determine the constant of proportionality.

Key Concepts

Laws of Logarithms
Graphing Functions

Question 4c

Marks: 5

c)

(i)

Show that, according to the model, it will take exactly

\frac{\ln 20}{\ln 7 - \ln 5}

hours (from

t = 0

) for the population of bacteria to grow to

100 000

.

(ii)

Confirm your answer to part (c)(i) graphically.

Key Concepts

The Logistic Equation
Separation of Variables

Question 5a

Marks: 8

After clearing a large forest of malign influences, a wizard introduces a population of 100 unicorns to the forest. According to the wizard’s mathemagicians, the population of unicorns in the forest may be modelled by the logistic equation

$\frac{d P}{d t} = 0.0006 P (250 - P)$

where $t$ is the time in years after the unicorns were introduced to the forest.

a)

Show that the population of unicorns at time

t

years is given by

$P (t) = \frac{500 e^{0.15 t}}{3 + 2 e^{0.15 t}}$

Key Concepts

Modelling with Differential Equations

Question 5b

Marks: 3

b)

Find the length of time predicted by the model for the population of unicorns to double in size.

Key Concepts

Limits
Modelling with Differential Equations

Question 5c

Marks: 2

c)

Determine the maximum size that the model predicts the population of unicorns can grow to.

Key Concepts

Homogeneous Differential Equations

Question 6a

Marks: 2

a)

Show that

$x^{2} \frac{d y}{d x} = x y + 2 x^{2}$

is a homogeneous differential equation.

Key Concepts

Homogeneous Differential Equations

Question 6b

Marks: 4

b)

Using the substitution

v = \frac{y}{x}

, show that the solution to the differential equation in part (a) is

$y = 2 x \ln |x| + c x$

where

c

is a constant of integration.

Key Concepts

Homogeneous Differential Equations
Implicit Differentiation

Question 7a

Marks: 3

a)

Use the substitution

v = \frac{y}{x}

to show that the differential equation

$y^{'} = \frac{y^{2}}{x^{2}} - \frac{y}{x} + 1$

may be rewritten in the form

$v^{'} = \frac{{(v - 1)}^{2}}{x}$

Key Concepts

Homogeneous Differential Equations
Implicit Differentiation

Question 7b

Marks: 5

b)

Hence use separation of variables to solve the differential equation in part (a) for which satisfies the boundary condition

y (1) = \frac{2}{3}

. Give your answer in the form

y = f (x)

.

Key Concepts

Separation of Variables
Finding the Constant of Integration

Question 8a

Marks: 2

Consider the differential equation

$y^{'} + 2 x y = (4 x + 2) e^{x}$

a)

Explain why it would be appropriate to use an integrating factor in attempting to solve the differential equation.

Key Concepts

Integrating Factor

Question 8b

Marks: 2

b)

Show that the integrating factor for this differential equation is

e^{x^{2}}

.

Key Concepts

Integrating Factor

Question 8c

Marks: 5

c)

Hence solve the differential equation.

Key Concepts

Integrating Factor
Reverse Chain Rule

Question 9

Marks: 7

Use an integrating factor to solve the differential equation

$(x + 3) \frac{d y}{d x} - 4 y = {(x + 3)}^{6}$

for $y$ which satisfies the boundary condition $y (- 2) = 0$ .

Key Concepts

Integrating Factor
Finding the Constant of Integration

Question 10a

Marks: 3

Consider the differential equation

$\frac{d y}{d x} = \frac{y}{x} + 1$

with the boundary condition $y (1) = 0$ .

a)

Apply Euler’s method with a step size of

h = 0.2

to approximate the solution to the differential equation at

x = 2

.

Key Concepts

Euler's Method: First Order

Question 10b

Marks: 4

b)

(i)

Explain what method you could use to solve the above differential equation analytically (i.e., exactly).

(ii)

The exact solution to the differential equation with the given boundary condition is

y = x \ln x

. Compare your approximation from part (a) to the exact value of the solution at

x = 2

.

Key Concepts

Homogeneous Differential Equations

Question 10c

Marks: 1

c)

Explain how the accuracy of the approximation in part (a) could be improved.

Key Concepts

Euler's Method: First Order

Question 11a

Marks: 3

A particle moves in a straight line, such that its displacement $x$ at time $t$ is described by the differential equation

$\frac{d x}{d t} = \frac{t e^{3 t^{2}} + 1}{4 x^{2}}, t \geq 0$

At time $t = 0$ , $x = \frac{1}{2}$ .

(a) By using Euler’s method with a step length of 0.1, find an approximate value for $x$ at time $t = 0.3$ .

Key Concepts

Euler's Method: First Order

Question 11b

Marks: 5

(b)

(i)

Solve the differential equation with the given boundary condition to show that

$x = \frac{1}{2} \sqrt[s]{e^{3 t^{2}} + 6 t}$

(ii)

Hence find the percentage error in your approximation for

x

at time

t = 0.3

.

Key Concepts

Separation of Variables
Reverse Chain Rule

DP IB Maths: AA HL

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5.10 Differential Equations

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