What is a phase portrait ?

A phase portrait is a diagram showing the trajectories of solutions over time for different initial conditions .

True or False? If the imaginary parts of complex eigenvalues are positive , then the solution trajectories move away from the origin .

False. If the real parts of complex eigenvalues are positive , then the solution trajectories (spirals) move away from the origin .

What is a stable equilibrium point ?

A stable equilibrium point is an equilibrium point where all solution trajectories close to the equilibrium point move towards the equilibrium point.

IBMathsDPApplications & Interpretation (AI)HLFlashcards5. CalculusCoupled & Second Order Differential Equations

Coupled & Second Order Differential Equations (DP IB Applications & Interpretation (AI))

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FrontCoupled Differential Equations
$\frac{d x}{d t} = a x + b y$
$\frac{d y}{d t} = c x + d y$
What is the matrix equation that corresponds to the coupled differential equations above?

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$\frac{d x}{d t} = a x + b y$
$\frac{d y}{d t} = c x + d y$
What is the matrix equation that corresponds to the coupled differential equations above?
$\frac{d x}{d t} = a x + b y$
$\frac{d y}{d t} = c x + d y$
The matrix equation that corresponds to the coupled differential equations above is $(\begin{matrix} \dot{x} \\ \dot{y} \end{matrix}) = (\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} x \\ y \end{matrix})$ where $\dot{x} = \frac{d x}{d t}$ and $\dot{y} = \frac{d y}{d t}$ . This is commonly written as $\dot{x} = M x$ .
What is a phase portrait?
A phase portrait is a diagram showing the trajectories of solutions over time for different initial conditions.
If the matrix $M$ has distinct, real, non-zero eigenvalues, $λ_{1}$ and $λ_{2}$ , and $p_{1}$ and $p_{2}$ are corresponding eigenvectors, then what is the general solution to $\dot{x} = M x$ ?
If the matrix $M$ has distinct, real, non-zero eigenvalues, $λ_{1}$ and $λ_{2}$ , and $p_{1}$ and $p_{2}$ are corresponding eigenvectors, then the general solution to $\dot{x} = M x$ is $x = A e^{λ_{1} t} p_{1} + B e^{λ_{2} t} p_{2}$ .
This is given in the formula booklet.
True or False?
If the matrix $M$ has distinct, real, non-zero eigenvalues, $λ_{1}$ and $λ_{2}$ , where $λ_{1} > λ_{2}$ , then the solution trajectories are approximately parallel to an eigenvector corresponding to $λ_{1}$ as $t$ tends to infinity.
True.
If the matrix $M$ has distinct, real, non-zero eigenvalues, $λ_{1}$ and $λ_{2}$ , where $λ_{1} > λ_{2}$ , then the solution trajectories are approximately parallel to an eigenvector corresponding to $λ_{1}$ as $t$ tends to infinity.
The eigenvector corresponding to the larger eigenvalue has a bigger effect for larger values of $t$ .
The matrix $M$ has distinct, positive eigenvalues, $λ_{1}$ and $λ_{2}$ , where $λ_{1} > λ_{2}$ , and $p_{1}$ and $p_{2}$ are corresponding eigenvectors. Describe the phase portrait for the solutions to $\dot{x} = M x$ .
If both eigenvalues are positive with $λ_{1} > λ_{2}$ , then all the solution trajectories move away from the origin as t increases. Initially, the trajectories are approximately parallel to $p_{2}$ and as $t$ gets large they are approximately parallel to $p_{1}$ .
The matrix $M$ has distinct, negative eigenvalues, $λ_{1}$ and $λ_{2}$ , where $λ_{1} > λ_{2}$ , and $p_{1}$ and $p_{2}$ are corresponding eigenvectors. Describe the phase portrait for the solutions to $\dot{x} = M x$ .
If both eigenvalues are negative with $λ_{1} > λ_{2}$ , then all the solution trajectories move towards the origin as t increases. Initially, the trajectories are approximately parallel to $p_{2}$ and as $t$ gets large they are approximately parallel to $p_{1}$ .
The matrix $M$ has distinct, real, eigenvalues, $λ_{1}$ and $λ_{2}$ , where $λ_{1} > 0 > λ_{2}$ , and $p_{1}$ and $p_{2}$ are corresponding eigenvectors. Describe the phase portrait for the solutions to $\dot{x} = M x$ .
If both eigenvalues have different signs then all the solution trajectories move towards the origin and then move away from the origin as $t$ increases. The origin is called a saddle point. Initially, if $λ_{1} > 0 > λ_{2}$ , the trajectories are approximately parallel to $p_{2}$ and as $t$ gets large they are approximately parallel to $p_{1}$ .
The matrix $M$ has imaginary eigenvalues. Describe the phase portrait for the solutions to $\dot{x} = M x$ .
If the eigenvalues are imaginary then all the solution trajectories are circles or ellipses with centres at the origin.
If the solution trajectories form circles, ellipses or spirals, how can you determine whether they move clockwise or anticlockwise?
If the solution trajectories form circles, ellipses or spirals, then you can determine whether they move clockwise or anticlockwise by finding the value of $\frac{d x}{d t}$ at a point on the $y$ -axis or the value of $\frac{d y}{d t}$ at a point on the $x$ -axis.
For example, if $\dot{y} < 0$ at (1, 0) or if if $\dot{x} > 0$ at (0, 1) then the motion is clockwise.
The matrix $M$ has two non-real eigenvalues that are complex conjugates with non-zero real parts. Describe the phase portrait for the solutions to $\dot{x} = M x$ .
If the eigenvalues are complex conjugates then all the solution trajectories are spirals.
True or False?
If the imaginary parts of complex eigenvalues are positive, then the solution trajectories move away from the origin.
False.
If the real parts of complex eigenvalues are positive, then the solution trajectories (spirals) move away from the origin.
In a phase portrait for the solutions to $\dot{x} = M x$ , what can you say about the eigenvalues if the solutions are moving away from the origin for all values of $t$ ?
In a phase portrait for the solutions to $\dot{x} = M x$ , if the solutions are moving away from the origin for all values of $t$ , then the eigenvalues are distinct and the real parts of both are positive.
This includes the case where they are both real and positive.
What is an equilibrium point of a solution to a coupled differential equation?
An equilibrium point of a solution to a coupled differential equation is a point where both $\frac{d x}{d t} = 0$ and $\frac{d y}{d t} = 0$ .
What is a stable equilibrium point?
A stable equilibrium point is an equilibrium point where all solution trajectories close to the equilibrium point move towards the equilibrium point.
What is a saddle point?
A saddle point is an equilibrium point where all nearby solution trajectories move towards the saddle point and then turn and move away from it.
For example, the origin is a saddle point on the following phase portrait.
If the solutions to $\dot{x} = M x$ have stable equilibrium points, then what can you say about the eigenvalues of $M$ ?
If the solutions to $\dot{x} = M x$ have stable equilibrium points, then the eigenvalues of $M$ are distinct and the real parts are negative.
This includes the case where both are real and negative.
If the solutions to $\dot{x} = M x$ have saddle points, then what can you say about the eigenvalues of $M$ ?
If the solutions to $\dot{x} = M x$ have saddle points, then the eigenvalues of $M$ are both real with different signs.
What is a second order differential equation?
A second order differential equation is a differential equation containing one or more second derivatives.
For example, $\frac{d^{2} x}{d t^{2}} + \frac{d x}{d t} + t = x^{2}$ is a second order differential equation.
How can you rewrite $\frac{d^{2} x}{d t^{2}} = f (x, \frac{d x}{d t}, t)$ as a coupled differential equation?
You can rewrite $\frac{d^{2} x}{d t^{2}} = f (x, \frac{d x}{d t}, t)$ as a coupled differential equation by:
- letting $y = \frac{d x}{d t}$ (which means $\frac{d y}{d t} = \frac{d^{2} x}{d t^{2}}$ ).
- Then $\frac{d x}{d t} = y$ and $\frac{d y}{d t} = f (x, y, t)$ .
True or False?
Euler's method can be used to find approximate solutions to $\frac{d^{2} x}{d t^{2}} = f (x, \frac{d x}{d t}, t)$ .
True.
Euler's method can be used to find approximate solutions to $\frac{d^{2} x}{d t^{2}} = f (x, \frac{d x}{d t}, t)$ .
You first need to write it as a pair of coupled differential equations.
How can you find the exact solutions to the second order differential equation $\frac{d^{2} x}{d t^{2}} + a \frac{d x}{d t} + b x = 0$ ?
To find the exact solutions to the second order differential equation $\frac{d^{2} x}{d t^{2}} + a \frac{d x}{d t} + b x = 0$ :
- Let $y = \frac{d x}{d t}$ and write as a coupled differential equation with $\frac{d y}{d t} = - b x - a y$ .
- Write as a matrix equation $(\begin{matrix} \dot{x} \\ \dot{y} \end{matrix}) = (\begin{matrix} 0 & 1 \\ - b & - a \end{matrix}) (\begin{matrix} x \\ y \end{matrix})$ .
- Find the general solution by finding the eigenvalues and corresponding eigenvectors of the matrix of coefficients.
- The solution to the original equation will be the top component (i.e. the $x$ component) of this vector.

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