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CONCORDIA UNIVERSITY / Engineering / ENGR 213 / What are differential equations?

What are differential equations?

What are differential equations?

Description

School: Concordia University
Department: Engineering
Course: Applied Ordinary Differential Equations
Term: Fall 2019
Tags: ENGR, 213, applied, Mathematics, Advanced, study, and guide
Cost: 50
Name: Study guide midterm ENGR213
Description: Chapters for the midterm
Uploaded: 10/19/2019
13 Pages 31 Views 2 Unlocks
Reviews


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"}.lst-kix_to0jk1m2pewh-1 > li:before{content:"- "}.lst-kix_to0jk1m2pewh-3 > li:before{content:"- "}.lst-kix_8cd53ccpz8c6-6 > li:before{content:"● "}.lst-kix_h6i3urvsyts3-6 > li:before{content:"" counter(lst-ctn-kix_h6i3urvsyts3-6,decimal) ". "}.lst-kix_to0jk1m2pewh-0 > li:before{content:"- "}.lst-kix_to0jk1m2pewh-4 > li:before{content:"- "}.lst-kix_8cd53ccpz8c6-7 > li:before{content:"○ "}.lst-kix_h6i3urvsyts3-8 > li:before{content:"" counter(lst-ctn-kix_h6i3urvsyts3-8,lower-roman) ". "}.lst-kix_8cd53ccpz8c6-8 > li:before{content:"■ "}.lst-kix_h6i3urvsyts3-7 > li:before{content:"" counter(lst-ctn-kix_h6i3urvsyts3-7,lower-latin) ". "}.lst-kix_to0jk1m2pewh-2 > li:before{content:"- "}ol.lst-kix_o7y7rq4si9gt-2{list-style-type:none}ol.lst-kix_o7y7rq4si9gt-3{list-style-type:none}ol.lst-kix_o7y7rq4si9gt-4{list-style-type:none}ol.lst-kix_o7y7rq4si9gt-5{list-style-type:none}ol.lst-kix_o7y7rq4si9gt-0{list-style-type:none}ol.lst-kix_o7y7rq4si9gt-1{list-style-type:none}.lst-kix_o7y7rq4si9gt-7 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-7}.lst-kix_8cd53ccpz8c6-5 > li:before{content:"■ "}.lst-kix_8cd53ccpz8c6-4 > li:before{content:"○ "}.lst-kix_hb8zkmnd809f-8 > li{counter-increment:lst-ctn-kix_hb8zkmnd809f-8}.lst-kix_hb8zkmnd809f-5 > li{counter-increment:lst-ctn-kix_hb8zkmnd809f-5}.lst-kix_8cd53ccpz8c6-3 > li:before{content:"● "}.lst-kix_8cd53ccpz8c6-0 > li:before{content:"● "}.lst-kix_8cd53ccpz8c6-2 > li:before{content:"■ "}ol.lst-kix_o7y7rq4si9gt-4.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-4 0}.lst-kix_8cd53ccpz8c6-1 > li:before{content:"○ "}.lst-kix_h6i3urvsyts3-2 > li{counter-increment:lst-ctn-kix_h6i3urvsyts3-2}.lst-kix_hb8zkmnd809f-2 > li:before{content:"" counter(lst-ctn-kix_hb8zkmnd809f-2,lower-roman) ". "}ul.lst-kix_8cd53ccpz8c6-5{list-style-type:none}ul.lst-kix_8cd53ccpz8c6-6{list-style-type:none}ul.lst-kix_8cd53ccpz8c6-3{list-style-type:none}ul.lst-kix_8cd53ccpz8c6-4{list-style-type:none}ul.lst-kix_8cd53ccpz8c6-1{list-style-type:none}ul.lst-kix_8cd53ccpz8c6-2{list-style-type:none}ul.lst-kix_8cd53ccpz8c6-0{list-style-type:none}.lst-kix_hb8zkmnd809f-0 > li:before{content:"" counter(lst-ctn-kix_hb8zkmnd809f-0,decimal) ". "}.lst-kix_hb8zkmnd809f-8 > li:before{content:"" counter(lst-ctn-kix_hb8zkmnd809f-8,lower-roman) ". "}.lst-kix_ojw8ousktcpl-1 > li{counter-increment:lst-ctn-kix_ojw8ousktcpl-1}ol.lst-kix_hb8zkmnd809f-5.start{counter-reset:lst-ctn-kix_hb8zkmnd809f-5 0}ul.lst-kix_8cd53ccpz8c6-7{list-style-type:none}ol.lst-kix_ojw8ousktcpl-8.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-8 0}ul.lst-kix_8cd53ccpz8c6-8{list-style-type:none}.lst-kix_hb8zkmnd809f-6 > li:before{content:"" counter(lst-ctn-kix_hb8zkmnd809f-6,decimal) ". "}ol.lst-kix_o7y7rq4si9gt-0.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-0 0}ol.lst-kix_o7y7rq4si9gt-3.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-3 0}ol.lst-kix_h6i3urvsyts3-2.start{counter-reset:lst-ctn-kix_h6i3urvsyts3-2 0}.lst-kix_h6i3urvsyts3-1 > li{counter-increment:lst-ctn-kix_h6i3urvsyts3-1}.lst-kix_hb8zkmnd809f-4 > li:before{content:"" counter(lst-ctn-kix_hb8zkmnd809f-4,lower-latin) ". "}.lst-kix_hb8zkmnd809f-4 > li{counter-increment:lst-ctn-kix_hb8zkmnd809f-4}.lst-kix_h6i3urvsyts3-0 > li{counter-increment:lst-ctn-kix_h6i3urvsyts3-0}ol.lst-kix_ojw8ousktcpl-8{list-style-type:none}.lst-kix_h6i3urvsyts3-1 > li:before{content:"" counter(lst-ctn-kix_h6i3urvsyts3-1,lower-latin) ". "}.lst-kix_ojw8ousktcpl-3 > li{counter-increment:lst-ctn-kix_ojw8ousktcpl-3}ol.lst-kix_ojw8ousktcpl-7{list-style-type:none}ol.lst-kix_ojw8ousktcpl-6{list-style-type:none}ol.lst-kix_ojw8ousktcpl-5{list-style-type:none}ol.lst-kix_ojw8ousktcpl-4{list-style-type:none}ol.lst-kix_ojw8ousktcpl-3{list-style-type:none}ol.lst-kix_ojw8ousktcpl-2{list-style-type:none}ol.lst-kix_ojw8ousktcpl-1{list-style-type:none}.lst-kix_ojw8ousktcpl-7 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-7,lower-latin) ". "}.lst-kix_it5xga617b9d-1 > li:before{content:"- "}ol.lst-kix_ojw8ousktcpl-0{list-style-type:none}.lst-kix_o7y7rq4si9gt-5 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-5}ol.lst-kix_h6i3urvsyts3-1.start{counter-reset:lst-ctn-kix_h6i3urvsyts3-1 0}.lst-kix_ojw8ousktcpl-3 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-3,decimal) ". "}.lst-kix_o7y7rq4si9gt-7 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-7,lower-latin) ". "}.lst-kix_ojw8ousktcpl-5 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-5,lower-roman) ". "}.lst-kix_it5xga617b9d-7 > li:before{content:"- "}.lst-kix_o7y7rq4si9gt-0 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-0}.lst-kix_o7y7rq4si9gt-1 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-1,lower-latin) ". "}.lst-kix_o7y7rq4si9gt-3 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-3,decimal) ". "}.lst-kix_o7y7rq4si9gt-6 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-6}.lst-kix_it5xga617b9d-5 > li:before{content:"- "}ol.lst-kix_hb8zkmnd809f-3.start{counter-reset:lst-ctn-kix_hb8zkmnd809f-3 0}.lst-kix_hb8zkmnd809f-3 > li{counter-increment:lst-ctn-kix_hb8zkmnd809f-3}.lst-kix_o7y7rq4si9gt-5 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-5,lower-roman) ". "}.lst-kix_it5xga617b9d-3 > li:before{content:"- "}.lst-kix_ojw8ousktcpl-1 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-1,lower-latin) ". "}.lst-kix_ojw8ousktcpl-8 > li{counter-increment:lst-ctn-kix_ojw8ousktcpl-8}ol.lst-kix_hb8zkmnd809f-2.start{counter-reset:lst-ctn-kix_hb8zkmnd809f-2 0}.lst-kix_ojw8ousktcpl-2 > li{counter-increment:lst-ctn-kix_ojw8ousktcpl-2}ol.lst-kix_o7y7rq4si9gt-1.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-1 0}.lst-kix_o7y7rq4si9gt-8 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-8}ol.lst-kix_hb8zkmnd809f-1.start{counter-reset:lst-ctn-kix_hb8zkmnd809f-1 0}.lst-kix_hb8zkmnd809f-0 > li{counter-increment:lst-ctn-kix_hb8zkmnd809f-0}.lst-kix_h6i3urvsyts3-7 > li{counter-increment:lst-ctn-kix_h6i3urvsyts3-7}ol.lst-kix_ojw8ousktcpl-1.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-1 0}ul.lst-kix_it5xga617b9d-8{list-style-type:none}.lst-kix_ojw8ousktcpl-5 > li{counter-increment:lst-ctn-kix_ojw8ousktcpl-5}ul.lst-kix_it5xga617b9d-0{list-style-type:none}ul.lst-kix_it5xga617b9d-1{list-style-type:none}ul.lst-kix_it5xga617b9d-2{list-style-type:none}ol.lst-kix_h6i3urvsyts3-0.start{counter-reset:lst-ctn-kix_h6i3urvsyts3-0 0}ul.lst-kix_it5xga617b9d-3{list-style-type:none}ul.lst-kix_it5xga617b9d-4{list-style-type:none}ul.lst-kix_it5xga617b9d-5{list-style-type:none}ul.lst-kix_it5xga617b9d-6{list-style-type:none}ul.lst-kix_it5xga617b9d-7{list-style-type:none}ol.lst-kix_hb8zkmnd809f-7.start{counter-reset:lst-ctn-kix_hb8zkmnd809f-7 0}ul.lst-kix_to0jk1m2pewh-4{list-style-type:none}ul.lst-kix_to0jk1m2pewh-3{list-style-type:none}ul.lst-kix_to0jk1m2pewh-6{list-style-type:none}ol.lst-kix_o7y7rq4si9gt-2.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-2 0}ul.lst-kix_to0jk1m2pewh-5{list-style-type:none}ul.lst-kix_to0jk1m2pewh-8{list-style-type:none}ul.lst-kix_to0jk1m2pewh-7{list-style-type:none}ul.lst-kix_to0jk1m2pewh-0{list-style-type:none}ul.lst-kix_to0jk1m2pewh-2{list-style-type:none}ol.lst-kix_ojw8ousktcpl-7.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-7 0}ol.lst-kix_h6i3urvsyts3-6.start{counter-reset:lst-ctn-kix_h6i3urvsyts3-6 0}ul.lst-kix_to0jk1m2pewh-1{list-style-type:none}ol.lst-kix_hb8zkmnd809f-6.start{counter-reset:lst-ctn-kix_hb8zkmnd809f-6 0}ol.lst-kix_ojw8ousktcpl-0.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-0 0}.lst-kix_h6i3urvsyts3-5 > li{counter-increment:lst-ctn-kix_h6i3urvsyts3-5}ol.lst-kix_ojw8ousktcpl-6.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-6 0}ol.lst-kix_hb8zkmnd809f-0.start{counter-reset:lst-ctn-kix_hb8zkmnd809f-0 0}ol.lst-kix_h6i3urvsyts3-5.start{counter-reset:lst-ctn-kix_h6i3urvsyts3-5 0}.lst-kix_o7y7rq4si9gt-1 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-1}.lst-kix_o7y7rq4si9gt-4 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-4}.lst-kix_hb8zkmnd809f-2 > li{counter-increment:lst-ctn-kix_hb8zkmnd809f-2}.lst-kix_ojw8ousktcpl-7 > li{counter-increment:lst-ctn-kix_ojw8ousktcpl-7}ol.lst-kix_hb8zkmnd809f-8{list-style-type:none}ol.lst-kix_hb8zkmnd809f-7{list-style-type:none}ol.lst-kix_hb8zkmnd809f-6{list-style-type:none}ol.lst-kix_hb8zkmnd809f-5{list-style-type:none}ol.lst-kix_hb8zkmnd809f-4{list-style-type:none}ol.lst-kix_hb8zkmnd809f-3{list-style-type:none}ol.lst-kix_hb8zkmnd809f-2{list-style-type:none}ol.lst-kix_hb8zkmnd809f-1{list-style-type:none}ol.lst-kix_hb8zkmnd809f-0{list-style-type:none}.lst-kix_hb8zkmnd809f-1 > li:before{content:"" counter(lst-ctn-kix_hb8zkmnd809f-1,lower-latin) ". "}.lst-kix_hb8zkmnd809f-3 > li:before{content:"" counter(lst-ctn-kix_hb8zkmnd809f-3,decimal) ". "}.lst-kix_o7y7rq4si9gt-2 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-2}.lst-kix_hb8zkmnd809f-7 > li:before{content:"" counter(lst-ctn-kix_hb8zkmnd809f-7,lower-latin) ". "}ol.lst-kix_ojw8ousktcpl-5.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-5 0}.lst-kix_hb8zkmnd809f-6 > li{counter-increment:lst-ctn-kix_hb8zkmnd809f-6}.lst-kix_hb8zkmnd809f-5 > li:before{content:"" counter(lst-ctn-kix_hb8zkmnd809f-5,lower-roman) ". "}.lst-kix_o7y7rq4si9gt-3 > li{counter-increment:lst-ctn-kix_o7y7rq4si9gt-3}ol.lst-kix_hb8zkmnd809f-8.start{counter-reset:lst-ctn-kix_hb8zkmnd809f-8 0}.lst-kix_h6i3urvsyts3-3 > li{counter-increment:lst-ctn-kix_h6i3urvsyts3-3}.lst-kix_hb8zkmnd809f-7 > li{counter-increment:lst-ctn-kix_hb8zkmnd809f-7}ol.lst-kix_o7y7rq4si9gt-6.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-6 0}ol.lst-kix_h6i3urvsyts3-5{list-style-type:none}ol.lst-kix_h6i3urvsyts3-4{list-style-type:none}ol.lst-kix_h6i3urvsyts3-3{list-style-type:none}ol.lst-kix_h6i3urvsyts3-2{list-style-type:none}.lst-kix_h6i3urvsyts3-2 > li:before{content:"" counter(lst-ctn-kix_h6i3urvsyts3-2,lower-roman) ". "}ol.lst-kix_h6i3urvsyts3-8{list-style-type:none}ol.lst-kix_h6i3urvsyts3-7{list-style-type:none}ol.lst-kix_h6i3urvsyts3-6{list-style-type:none}.lst-kix_ojw8ousktcpl-0 > li{counter-increment:lst-ctn-kix_ojw8ousktcpl-0}.lst-kix_h6i3urvsyts3-0 > li:before{content:"" counter(lst-ctn-kix_h6i3urvsyts3-0,decimal) ". "}ol.lst-kix_h6i3urvsyts3-1{list-style-type:none}ol.lst-kix_h6i3urvsyts3-0{list-style-type:none}.lst-kix_ojw8ousktcpl-6 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-6,decimal) ". "}.lst-kix_ojw8ousktcpl-8 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-8,lower-roman) ". "}.lst-kix_it5xga617b9d-2 > li:before{content:"- "}ol.lst-kix_h6i3urvsyts3-8.start{counter-reset:lst-ctn-kix_h6i3urvsyts3-8 0}.lst-kix_ojw8ousktcpl-4 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-4,lower-latin) ". "}.lst-kix_o7y7rq4si9gt-8 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-8,lower-roman) ". "}.lst-kix_it5xga617b9d-0 > li:before{content:"- "}.lst-kix_it5xga617b9d-8 > li:before{content:"- "}.lst-kix_o7y7rq4si9gt-2 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-2,lower-roman) ". "}.lst-kix_ojw8ousktcpl-0 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-0,decimal) ". "}.lst-kix_it5xga617b9d-6 > li:before{content:"- "}.lst-kix_o7y7rq4si9gt-6 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-6,decimal) ". "}.lst-kix_ojw8ousktcpl-2 > li:before{content:"" counter(lst-ctn-kix_ojw8ousktcpl-2,lower-roman) ". "}ol.lst-kix_ojw8ousktcpl-3.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-3 0}.lst-kix_o7y7rq4si9gt-4 > li:before{content:"" counter(lst-ctn-kix_o7y7rq4si9gt-4,lower-latin) ". "}.lst-kix_h6i3urvsyts3-4 > li{counter-increment:lst-ctn-kix_h6i3urvsyts3-4}ol.lst-kix_o7y7rq4si9gt-8.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-8 0}.lst-kix_it5xga617b9d-4 > li:before{content:"- "}ol.lst-kix_h6i3urvsyts3-7.start{counter-reset:lst-ctn-kix_h6i3urvsyts3-7 0}ol.lst-kix_ojw8ousktcpl-2.start{counter-reset:lst-ctn-kix_ojw8ousktcpl-2 0}ol.lst-kix_o7y7rq4si9gt-7.start{counter-reset:lst-ctn-kix_o7y7rq4si9gt-7 0}

Initial Value Problem        (IVP)If you want to learn more check out What is the equation of homogenous?

1.2 initial value problems

can be considered in many ways

  1. As a function, a function of all real numbers X except X=
  2. To a solution of the DE , on the interval I.

→ x cant be equal to  because the function cant be defined and differentiate.

  1. As a solution of the IVP ; y(0) = -1, on the initial I, (x ) the interval also needs to contain the initial point X=0

Don't forget about the age old question of What are the challenges that arise being a manager?

IVP: its a solution of a DE that satisfies initial conditionsWe also discuss several other topics like Describe the functional divisions of the nervous system.

Steps:

  1. We find the solution of the DE
  2. We plug in the initial conditions to find C()
  3. We rewrite the solution for the IVP with the value of C

1.3 Differential equation (DEs) as mathematical modelsWe also discuss several other topics like How do you arrange the elements according to properties?

The mathematical description of a system or a phenomenon is growth, radioactivity, heating,....

  1. first , wee have to read the situation and identify the variables
  2. Make a set of reasonable assumptions about the system

→ they include any applicable laws to the system.If you want to learn more check out How do you think like a scientist?

→ the assumptions will include a note of change (frequently)

  1. Find the mathematical model
  2. Solve the model

        If the results are weak, we increase the level of resolution:We also discuss several other topics like Differentiate competitive and noncompetitive inhibitors of enzymes.

Assumptions and hypotheses → express assumptions in terms of DEs → mathematical formation → solve the DE → obtain solution → display predictions of the model(graphically) → check model predictions with known facts → necessary, alter assumption or increase resolution of the model.

  • Of course, by increasing the resolution we add to the complexity of the mathematical model and increase the chances of not not finding an explicit solution.
  • In a physical system, the variable t will often be involved

→ the dependent variable(s) describe the system in the fast, preset or future.

Population dynamics.

  • The more people there is at time t, the more there going to be in the future.
  • In mathematical therms: (PLH denotes the total population at time t)

        ∝P or kP

Where K is a constant of proportionality.

Remark, this model however fail to take things like em/immigration into account

→ good for small population over short intervals (em: bacteria growing)

Radioactivity decay:

  • The rate DA/Dt at which a substance decays is proportional to the amount A(t) of the substance remaining at the time t:

Decrease → k(-)         ∝A or                 

        This is the equation as in the population growth.

  1. A=A(5) → A(0) = C
  2. Sample go from 100mg to 68mg

C=100                A=68

In 100 days, it decrease by 33%(66%left)        → 68=100

Conclusion for both equation: → it single differential equation can serve as a mathematical model for many different phenomena.

MM are often accompanied with side conditions.

→ ex: an initial population Po, if this population is at t = 0

        Then → P(0) = Po

Mathematical model → initial value problem(condition) IVP

                        → boundary - value problem(section 3.9)

Newton`s law of cooling / warming

∝.T-Tm        or =k(T-Tm)

Where T(t): represents the temp of a body at time t

        Tm : the temp of the surrounding medium.

        : the rate at which the temp. Of the body changes

        k : constant of proportionality (k<0 if Tm is constant)

Spread of disease

        

Where  x(t): denotes the number of people having the disease

        y(t): number of people who have not been yet exposed.

        : the rate of which the disease spreads

        → this rate is proportional to the number of interaction between the two groups (x(t) and y(t))

        So it is also proportional to the product of Xy

        K: constant of proportionality.

→ em: a community as a amount n of people.

        1 person with a disease is introduced in this community.

So → x(t) and y(t) are related by x + y = n + 1

=kx(n + 1 - x)

At exponential growth →

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