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? 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 .

Given a boundary condition, how do you sketch a solution curve using slope fields ?

Given a boundary condition, you can sketch a solution curve using slope fields by starting at the given point and using the tangent lines to help you determine the steepness of the solution curve changes.

True or False? A solution curve can cut across the tangent lines on a slope field diagram.

False. A solution curve should not cut across the tangent lines on a slope field diagram.

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.

IBMathsDPApplications & Interpretation (AI)HLFlashcards5. CalculusDifferential Equations

Differential Equations (DP IB Applications & Interpretation (AI))

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FrontModelling with Differential Equations
What is a differential equation?
FrontModelling with Differential Equations
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"?
BackModelling with Differential Equations
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)

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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.
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 $k > 0$ .
What method can be used to solve a differential equation of the form $\frac{d N}{d t} = k N$ , where $k$ is a constant?
A differential equation of the form $\frac{d N}{d t} = k N$ , where $k$ is a constant, can be solved using separation of variables.
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 slope field for the equation $\frac{d y}{d x} = g (x, y)$ ?
A slope field for the equation $\frac{d y}{d x} = g (x, y)$ is a diagram with short tangent lines drawn at a number of points. These are used to give an idea of what the graphs of the solutions look like.
For example, the slope field for $\frac{d y}{d x} = y \sin x - e^{- \cos x} \cos x$ is shown below.
True or False?
When drawing a solution on a slope field, you need to join the tangent lines together.
False.
When drawing a solution on a slope field, you should not join the tangent lines together.
An example is shown below.
How do you find the set of points for which the solutions to $\frac{d y}{d x} = g (x, y)$ will have horizontal tangents?
To find the set of points for which the solutions to $\frac{d y}{d x} = g (x, y)$ will have horizontal tangents, you solve the equation $g (x, y) = 0$ .
Given a boundary condition, how do you sketch a solution curve using slope fields?
Given a boundary condition, you can sketch a solution curve using slope fields by starting at the given point and using the tangent lines to help you determine the steepness of the solution curve changes.
True or False?
A solution curve can cut across the tangent lines on a slope field diagram.
False.
A solution curve should not cut across the tangent lines on a slope field diagram.
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.
State the recursion equations that are used when applying Euler's method to find an approximate solution for coupled differential equations of the form $\frac{d x}{d t} = f_{1} (x, y, t)$ and $\frac{d y}{d t} = f_{2} (x, y, t)$ .
The recursion equations are:
$x_{n + 1} = x_{n} + h \times f_{1} (x_{n}, y_{n}, t_{n})$
$y_{n + 1} = y_{n} + h \times f_{2} (x_{n}, y_{n}, t_{n})$
$t_{n + 1} = t_{n} + h$
Where:
- $h$ is the constant step length
These equations are given in the exam formula booklet.

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