What is Euler's method ?

Euler's method is a numerical method for finding approximate solutions to differential equations.

In general, how can the accuracy of Euler's method be improved?

The accuracy of Euler's method can be improved by decreasing the step size h .

True or False? Euler's method always gives exact solutions to differential equations.

False. Euler's method gives approximate solutions to differential equations.

What are boundary conditions in differential equations?

Boundary conditions are known values (usually initial values ) given for the variables in a differential equation.

True or False? All first order differential equations can be solved using separation of variables .

False. Only certain types of first order differential equations can be solved using separation of variables .

What is an integrating factor ?

An integrating factor is a function that both sides of a differential equation can be multiplied by to make the equation exactly integrable.

Why are differential equations useful for modelling real-world situations?

Differential equations are useful for modelling real-world situations because, like many real-world situations, they deal with rates of change and how variables change with respect to one another.

True or False? The logistic equation always results in population growth.

False. The logistic equation can result in population growth or decline depending on the values of k and a, and the size of the initial population.

What is the main advantage of using the logistic equation over simpler growth models?

The main advantage of using the logistic equation is that it incorporates limiting factors . These set, for example, a maximum size that a population might grow to, which provides a more realistic model for real-world populations.

What technique is typically used to solve a logistic equation ?

The technique of separation of variables is typically used to solve a logistic equation .

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Differential Equations (DP IB Analysis & Approaches (AA))

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What is a differential equation?

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What is a differential equation?
A differential equation is an equation that contains derivatives.
E.g. $\frac{d y}{d x} = 12 x y^{2}$ and $\frac{d^{2} x}{d t^{2}} - 5 \frac{d x}{d t} + 7 x = 5 \sin t$ are both differential equations.
What is a first order differential equation?
A first order differential equation is a differential equation that contains first derivatives but no second (or higher) derivatives.
E.g. $\frac{d y}{d x} = 12 x y^{2}$ is a first order differential equation, but $\frac{d^{2} x}{d t^{2}} - 5 \frac{d x}{d t} + 7 x = 5 \sin t$ is not (because of the $\frac{d^{2} x}{d t^{2}}$ term).
What is Euler's method?
Euler's method is a numerical method for finding approximate solutions to differential equations.
State the recursion equations that are used when applying Euler's method to find an approximate solution for a differential equation of the form $\frac{d y}{d x} = f (x, y)$ .
The recursion equations that are used when applying Euler's method to find an approximate solution for a differential equation of the form $\frac{d y}{d x} = f (x, y)$ are $y_{n + 1} = y_{n} + h \times f (x_{n}, y_{n})$ and $x_{n + 1} = x_{n} + h$
Where:
- $h$ is the constant step length
These equations are given in the exam formula booklet.
In general, how can the accuracy of Euler's method be improved?
The accuracy of Euler's method can be improved by decreasing the step size h.
True or False?
Euler's method always gives exact solutions to differential equations.
False.
Euler's method gives approximate solutions to differential equations.
What are boundary conditions in differential equations?
Boundary conditions are known values (usually initial values) given for the variables in a differential equation.
True or False?
All first order differential equations can be solved using separation of variables.
False.
Only certain types of first order differential equations can be solved using separation of variables.
What form must a differential equation be in to use separation of variables?
To use separation of variables, a differential equation must be in the form $\frac{d y}{d x} = g (x) h (y)$ .
E.g. $\frac{d y}{d x} = x y^{2}$ or $\frac{d y}{d x} = \frac{\sin x}{y^{3}} = \sin x (\frac{1}{y^{3}})$ .
What are the steps for solving a differential equation using separation of variables?
The steps for solving a differential equation using separation of variables are:
1. Rearrange $\frac{d y}{d x} = g (x) h (y)$ into the form $\frac{1}{h (y)} \frac{d y}{d x} = g (x)$ .
2. Integrate both sides with respect to $x$ to get $\int \frac{1}{h (y)} d y = \int g (x) d x$ .
3. Solve the integrals.
4. Use boundary or initial conditions (if any).
5. Rearrange (if necessary).
True or False?
The differential equation $\frac{d y}{d x} = 6 y$ cannot be solved using separation of variables, because there is no function of $x$ on the right-hand side.
False.
The differential equation in $\frac{d y}{d x} = 6 y$ can be solved using separation of variables.
In this case the 'function of $x$ ' on the right-hand side is the 6. I.e. let $g (x) = 6$ and let $h (y) = y$ , and then solve using separation of variables as usual.
What is a homogeneous first order differential equation?
A homogeneous first order differential equation is one that can be written in the form $\frac{d y}{d x} = f (\frac{y}{x})$ .
What substitution is used to solve homogeneous differential equations?
The substitution $v = \frac{y}{x}$ is used to solve homogeneous differential equations.
True or False?
By using the product rule and implicit differentiation, $\frac{d y}{d x}$ in a homogeneous differential equation can be replaced by the substitution $v - x \frac{d v}{d x}$ .
False.
By using the product rule and implicit differentiation, $\frac{d y}{d x}$ in a homogeneous differential equation can be replaced by the substitution $v + x \frac{d v}{d x}$ .
- Start with the standard substitution $v = \frac{y}{x} \Rightarrow y = x v$ .
- Differentiate both sides with respect to $x$ , i.e. $\frac{d y}{d x} = \frac{d}{d x} (v x)$ .
- Use implicit differentiation and the product rule, i.e. $\frac{d y}{d x} = v + x \frac{d v}{d x}$
What is the standard form for an integral that can be solved using an integrating factor?
The standard form for an integral that can be solved using an integrating factor is $y^{'} + P (x) y = Q (x)$ , where $y^{'} = \frac{d y}{d x}$ .
What is an integrating factor?
An integrating factor is a function that both sides of a differential equation can be multiplied by to make the equation exactly integrable.
What is the formula for the integrating factor?
The formula for the integrating factor to solve a differential equation of the form $y^{'} + P (x) y = Q (x)$ is $e^{\int P (x) d x}$ .
This formula is in the exam formula booklet.
What are the steps for solving a differential equation using an integrating factor?
The steps for solving a differential equation using an integrating factor are:
1. Rearrange into the standard form $y^{'} + P (x) y = Q (x)$ .
2. Integrate $P (x)$ to find the integrating factor $e^{\int P (x) d x}$ .
3. Multiply both sides of the equation by the integrating factor.
4. Integrate both sides.
5. Rearrange the solution into the form $y = f (x)$ .
Why are differential equations useful for modelling real-world situations?
Differential equations are useful for modelling real-world situations because, like many real-world situations, they deal with rates of change and how variables change with respect to one another.
What equation can be written down right away on the basis of the information that "the rate of change of a population, P, at a particular time is proportional to the size of the population at that time"?
The information "the rate of change of a population, P, at a particular time is proportional to the size of the population at that time"? is equivalent to the equation $\frac{d P}{d t} = k P$
Where:
- $k$ is the constant of proportionality (which will usually need to be found)
- $t$ is the variable for time (often, but not always, measured in seconds)
True or False?
The simple model $\frac{d N}{d t} = k N$ represents unlimited exponential growth when $k > 0$ .
True.
The simple model $\frac{d N}{d t} = k N$ represents unlimited exponential growth when $k > 0$ .
State the standard form of the logistic equation.
The standard form of the logistic equation is $\frac{d N}{d t} = k N (a - N)$
Where:
- t is the time (since the moment defined as $t = 0$ ) that the population has been growing
- N is the size of the population at time $t$
- $k$ is a constant determining the relative rate of population growth
- $a$ is a constant that places a limit on the maximum size to which the population can grow
This is not in the exam formula booklet. However the exact form of any logistic equation you need to use will always be given in an exam question.
True or False?
The logistic equation always results in population growth.
False.
The logistic equation can result in population growth or decline depending on the values of k and a, and the size of the initial population.
What is the main advantage of using the logistic equation over simpler growth models?
The main advantage of using the logistic equation is that it incorporates limiting factors. These set, for example, a maximum size that a population might grow to, which provides a more realistic model for real-world populations.
What technique is typically used to solve a logistic equation?
The technique of separation of variables is typically used to solve a logistic equation.

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Differential Equations (DP IB Analysis & Approaches (AA))

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1. Number & Algebra

Number & Algebra Toolkit

Exponentials & Logs

Sequences & Series

Simple Proof & Reasoning

Proof by Induction & Contradiction

Binomial Theorem

Permutations & Combinations

Complex Numbers

Further Complex Numbers

Systems of Linear Equations

2. Functions

Linear Functions & Graphs

Quadratic Functions & Graphs

Functions Toolkit

Other Functions & Graphs

Reciprocal & Rational Functions

Transformations of Graphs

Polynomial Functions

Inequalities

Modulus Functions & Further Transformations

3. Geometry & Trigonometry

Geometry Toolkit

Geometry of 3D Shapes

Trigonometry

The Unit Circle & Exact Values

Trigonometric Functions & Graphs

Trigonometric Equations & Identities

Inverse & Reciprocal Trigonometric Functions

Trigonometric Proof & Equation Strategies

Vector Properties

Vector Equations of Lines

Vector Planes

4. Statistics & Probability

Statistics Toolkit

Correlation & Regression

Probability

Discrete Random Variables

Binomial Distribution

Normal Distribution

Continuous Random Variables

5. Calculus

Differentiation

Techniques & Applications of Differentiation

Integration

Techniques & Applications of Integration

Optimisation

Kinematics

Basic Limits & Continuity

Further Differentiation

Further Integration

Differential Equations

Maclaurin Series

Limits using l'Hôpital's Rule & Maclaurin Series