5.10 Differential Equations

1 2a 2b 3a 3b 4a 4b 5a 5b 5c 5d 5e 6 7a 7b 8 9a 9b 9c 10a 10b 10c

Question 1

Marks: 5

Consider the first-order differential equation

$\frac{d y}{d x} + \frac{1}{2 x} = \sin 3 x \cos 3 x$

Solve the equation given that $y = 0$ when $x = \frac{π}{2}$ , giving your answer in the form $y = f (x)$ . .

Key Concepts

Double Angle Formulae
Finding the Constant of Integration

Question 2a

Marks: 4

Use separation of variables to solve each of the following differential equations

(a)

\frac{d y}{d x} = \frac{3 y^{4}}{4 x^{3}}

Key Concepts

Separation of Variables

Question 2b

Marks: 5

(b)

\frac{d y}{d x} = \frac{x^{2}}{y (π - x^{3})} e^{y^{2}}

Key Concepts

Separation of Variables

Question 3a

Marks: 5

Solve each of the following differential equations for $y$ which satisfies the given boundary condition, giving your answers in the form $y = f (x)$ .

(a)

\cos π x^{4} \frac{d y}{d x} = \tan π x^{4} {(\frac{x}{y})}^{3}; y (0) = - 3

Key Concepts

Separation of Variables
Integrating with Reciprocal Trigonometric Functions

Question 3b

Marks: 6

(b)

e^{x^{2}} \cos e c y \frac{d y}{d x} = x \sin y; y (0) = \frac{3 π}{4}

Key Concepts

Separation of Variables
Integrating with Inverse Trigonometric Functions

Question 4a

Marks: 8

As the atoms in a sample of radioactive material undergo radioactive decay, the rate of change of the number of radioactive atoms remaining in the sample at any time $t$ is proportional to the number, $N$ , of radioactive atoms currently remaining. The amount of time, $λ$ , that it takes for half the radioactive atoms in a sample of radioactive material to decay is known as the half-life of the material.

Let $N_{0}$ be the number of radioactive atoms originally present in a sample.

(a)

By first writing and solving an appropriate differential equation, show that the number of radioactive atoms remaining in the sample at any time

t \geq 0

may be expressed as

N (t) = N_{0} e^{- \frac{\ln 2}{λ} t}

Key Concepts

Modelling with Differential Equations
Separation of Variables

Question 4b

Marks: 3

Plutonium-239, a by-product of uranium fission reactors, has a half-life of 24000 years.

(b)

For a particular sample of Plutonium-239, determine how long it will take until less than 1% of the original radioactive Plutonium-239 atoms in the sample remain.

Key Concepts

Solving Inequalities Graphically

Question 5a

Marks: 2

Consider the standard logistic equation

$\frac{d P}{d t} = k P (a - P)$

where $P$ is the size of a population at time $t \geq 0$ , and where $k$ and $a$ are positive constants. Let the population at time $t = 0$ be denoted by $P_{0}$ .

(a)

Write down the solution to the logistic equation in the case where

P_{0} = a

, using mathematical reasoning to justify your answer.

Key Concepts

The Logistic Equation
Introduction to Derivatives

Question 5b

Marks: 8

(b)

In the case where

P_{0} \neq α

, show that the solution to the logistic equation is

P (t) = \frac{a A e^{a k t}}{1 + A e^{a k t}}

where

A

is an arbitrary constant.

Key Concepts

Separation of Variables
Integrating with Partial Fractions

Question 5c

Marks: 2

(c)

In the case where $P_{0} \neq α$ , write down an expression for $A$ in terms of $a$ and $P_{0}$ .

Question 5d

Marks: 3

(d)

In the case where

P_{0} \neq 0

, determine the behaviour of

P

t

becomes large.

Key Concepts

Limits
Modelling with Differential Equations

Question 5e

Marks: 4

(e)

In the case where

0 < 2 P_{0} < a

, determine the value of

t

at which the initial population will have doubled. Your answer should be given explicitly in terms of

a, k

and

P_{0}

Key Concepts

Modelling with Differential Equations
The Logistic Equation

Question 6

Marks: 8

Solve the differential equation

$x \frac{d y}{d x} - y = \frac{x y^{2}}{y^{2} \sin (\frac{y}{x}) - x^{2} \cos (\frac{x}{y})}$

Key Concepts

Homogeneous Differential Equations
Reverse Chain Rule

Question 7a

Marks: 9

Consider the differential equation

$x^{2} y' = y^{2} + 3 x y - 8 x^{2}$

with the boundary condition $y (1) = - 3$ .

(a)

Solve the differential equation for

y

which satisfies the given boundary condition, giving your answer in the form

y = f (x)

Key Concepts

Homogeneous Differential Equations
Integrating with Partial Fractions

Question 7b

Marks: 3

(b)

Determine the asymptotic behaviour of the graph of the solution as

x

becomes large.

Key Concepts

Limits
Key Features of Graphs

Question 8

Marks: 7

Solve the differential equation

$(4 x^{2} + 1) y' + y = \frac{1 - x + 4 x^{2} - 4 x^{3}}{\sqrt{e^{\arctan 2 x}}}$

Key Concepts

Integrating Factor

Question 9a

Marks: 3

Consider the differential equation

$\frac{d y}{d x} = \frac{5}{\sqrt{63 + 11 x^{2} - 2 x^{4}}} - \frac{2 x y}{2 x^{2} + 7}$

with the boundary condition $y (- \frac{3 \sqrt{2}}{2}) = 1$ .

(a)

Apply Euler’s method with a step size of

h = 0.2

to approximate the solution to the differential equation at

x = \frac{2 - 3 \sqrt{3}}{2}

Key Concepts

Euler's Method: First Order

Question 9b

Marks: 7

(b)

Solve the differential equation analytically, for

y

which satisfies the given boundary condition.

Key Concepts

Integrating Factor
Integrating with Inverse Trigonometric Functions

Question 9c

Marks: 3

(c)

(i)

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

x = \frac{2 - 3 \sqrt{2}}{2}

(ii)

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

Key Concepts

Euler's Method: First Order

Question 10a

Marks: 3

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

$\frac{d x}{d t} = \frac{\sin t}{1 + \cos^{2} t}, 0 \leq t \leq 3$

At time $t = 1.6, x = 1 .$

(a)

By using Euler’s method with a step length of 0.04, find an approximate value for

x

at time

t = 1.8

Key Concepts

Euler's Method: First Order
Displacement, Velocity & Acceleration

Question 10b

Marks: 7

The diagram below shows a graph of the exact solution $x = f (t)$ to the differential equation with the given boundary condition.

q10b_5-10_differential-equations_veryhard_ib_aa_hl_maths_diagram

Given that the graph of $x = f (t)$ has exactly one point of inflection, find the exact value of the $t$ -coordinate of the point of inflection.

Key Concepts

Points of Inflection

Question 10c

Marks: 3

(c)

Hence determine whether the approximation found in part (a) will be an overestimate or an underestimate for the true value of

x

when

t = 1.8

. Be sure to use mathematical reasoning to justify your answer.

Key Concepts

Euler's Method: First Order

DP IB Maths: AA HL

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