First order linear equations Applications of ODE What you will...
Transcript of First order linear equations Applications of ODE What you will...
First order linear equationsApplications of ODE
What you will learn today
First order linear equations
Applications of ODE
Differential Equations 1/24
First order linear equationsApplications of ODE
A first order linear differential equation is one that can be put intothe form,
dy
dx+ P(x)y = Q(x)
where P and Q are continuous functions on a given interval.
Example
y ′ + 1x y = 2 is a first order linear equation.
y ′ + xy2 = 2 is not.
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First order linear equationsApplications of ODE
How to solve this?
Example
y ′ +1
xy = 2
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First order linear equationsApplications of ODE
When an apple accidentally hit Newton’s head, he realizes that ifhe multiplies both sides of the equation by an extra factor x , theequation becomes
xy ′ + y = 2x
(...?)
Differential Equations 4/24
First order linear equationsApplications of ODE
Even more accidentally, he realizes that (xy)′ = xy ′ + y , therefore
d(xy)
dx= 2x
( or substitute z = xy if you like.)Integrate both sides, have∫
d(xy) =
∫2xdx
xy = x2 + C
y = x +C
x
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First order linear equationsApplications of ODE
Wait a minuet...How can we make these ”accidental observations”into a more general rule?Let’s see what happened in our example:Step 1, given the equation
y ′ + P(x)y = Q(x)
we would like to multiply an extra function I (x) (called integratingfactor) on both sides, s.t. the LHS becomes the derivative of somenew function, call it z(x).Step 2, solve the new equation
z ′ = I (x)Q(x)
Step 3, from z(x) get back y(x).
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First order linear equationsApplications of ODE
Guess: Could step 1 always be achieved for a FOLODE?After multiplying I (x), LHS becomes I (x)y ′ + I (x)P(x)y .Daringly want z(x) = J(x)y , z ′ = J(x)y ′ + J ′(x)y .Compare, we let J(x) = I (x), J ′(x) = I (x)P(x).Therefore I ′(x) = I (x)P(x), which is a separable ODE for I (x)itself!
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First order linear equationsApplications of ODE
Let’s solve for I (x) first.
I (x) = Ce∫P(x)dx
For simplicity can pick C=1.
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First order linear equationsApplications of ODE
Going back to the original problem, we have
z =
∫I (x)Q(x)dx
where z = I (x)y .Therefore y = 1
I (x)
∫I (x)Q(x)dx , where we picked
I (x) = e∫P(x)dx .
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First order linear equationsApplications of ODE
Theorem
For a FOLODE like y ′ + P(x)y = Q(x), the general solution isy = 1
I (x)
∫I (x)Q(x)dx, where I (x) = e
∫P(x)dx .
A sweet lie has been woven when someone gives you an equation
y ′ = P(x)y + Q(x)
. You can not apply the above formula directly! (why?)
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First order linear equationsApplications of ODE
Example
dy
dx+ 3x2y = 6x2
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First order linear equationsApplications of ODE
Example
x2y ′ + xy = 1, x > 0, y(1) = 2
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First order linear equationsApplications of ODE
Example
y ′ + 2xy = 1
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
Situation:t: timeE(t): voltage(V)I(t): current(A)R: resistance(Ω)L: inductance(H), a constantVoltage drop due to resistor is RI; due to inductor is LdI
dt .
LdI
dt+ RI = E (t)
where I is the unknown function, E (t) is a given function, this is aFOLODE for I .
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
Example
Solve the initial value problem L=4, R=12, E(t)=60, I(0)=0.What is the limiting value of the current?
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
Models for the growth of population,t: time, (variable)P: number of individuals in the population, (unknown function)For simplicity, let’s assume the rate of population growth isproportion to P:
dP
dt= kP
where k > 0 is a constant.
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
Solving, get
P = Cet
where C depends on the initial condition. This is a model forpopulation growth in ideal condition: sufficient food and resources,no epidemics...
-0.5 0.5 1.0 1.5
-10
-5
5
10
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
A more realistic model takes the resource limit into account: k isnot a constant.Rather, say the environmental resource limit is K, as P
K ⇒ 1,k ⇒ 0.i.e.
k = k1(1− P
K)
which is a separable equation. This model was called the logisticdifferential equation. It was proposed by Pierre-Francios Verhulst(Dutch) in the 1840s.
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
From the equation we can see when P = 0 or P = K , dPdt = 0,
therefore the population stays constant. These are calledequilibrium solutions.The general solution to the logistic model is
P(t) =K
1 + Ae−k1t
where A is a constant.
2 4 6 8 10
2000
4000
6000
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10 000
12 000
14 000
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
True or false: when there are no wolves, sheep would eat up all thegrass and die. Therefore the existence of wolves is beneficial to thelong term survival of the specie of sheep.
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
S: population of sheep W: wolves k, r , a, b are constants. On anice afternoon, sheep are wandering on the streets of Philly, whatis the chance that any sheep meets any wolf? It is proportional toboth the density of S and W , so under our postulate say it isproportional to SW .We have a coupled system of differential equations:
dS
dt= kS − aSW
dW
dt= −rW + bSW
This predator-prey system is called the Lotka-Volterra equations.Note the time t doesn’t show up on the RHS of either of theabove equations, also called autonomous.
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
Q: find the equilibrium solutions of the system.Eliminate t from our observation:
dS
dW=
kS − aSW
−rW + bSW
The S −W plane (without t) is called the phase plane, thesolution curves on the phase plane are called phase trajectories. Aphase portrait consists of equilibrium points and typical phasetrajectories.Q: In which direction is a point moving in a trajectory? How fast isit moving? Is the equilibrium solution stable? Is it asymptoticallystable?
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
Look at Prob 9 in §10.6
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First order linear equationsApplications of ODE
Electric circuitsPopulation growthPredator-Prey Systems
What you have learnt today:
First order linear equations
Model for electric circuit,
logistic model for population growth,
Lotka-Volterra model for predator-prey system.
Differential Equations 24/24